Theorem csbsngVD 44569

Description: Virtual deduction proof of csbsng 4717. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbsng 4717 is csbsngVD 44569 without virtual deductions and was automatically derived from csbsngVD 44569.

1::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
3:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑦 = 𝑦   )
4:3:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
5:2,4:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
6:5:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
7:6:	⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
8:1:	⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   )
9:7,8:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
10::	⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
11:10:	⊢ ∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
12:1,11:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = ⦋ 𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   )
13:9,12:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { 𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
14::	⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}
15:13,14:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { ⦋𝐴 / 𝑥⦌𝐵}   )
qed:15:	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋ 𝐴 / 𝑥⦌𝐵})

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
csbsngVD	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})

Proof of Theorem csbsngVD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn1 44250	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2		sbceqg 4414	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵))
3	1, 2	e1a 44303	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
4		csbconstg 3911	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
5	1, 4	e1a 44303	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑦 = 𝑦   )
6		eqeq1 2730	. . . . . . . . 9 ⊢ (⦋𝐴 / 𝑥⦌𝑦 = 𝑦 → (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵))
7	5, 6	e1a 44303	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
8		bibi1 350	. . . . . . . . 9 ⊢ (([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵) → (([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) ↔ (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)))
9	8	biimprd 247	. . . . . . . 8 ⊢ (([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵) → ((⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) → ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)))
10	3, 7, 9	e11 44364	. . . . . . 7 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
11	10	gen11 44292	. . . . . 6 ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
12		abbib 2798	. . . . . . 7 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} ↔ ∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵))
13	12	biimpri 227	. . . . . 6 ⊢ (∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵})
14	11, 13	e1a 44303	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
15		csbab 4442	. . . . . . . 8 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵}
16	15	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵})
17	16	eqcomd 2732	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵})
18	1, 17	e1a 44303	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   )
19		eqeq1 2730	. . . . . 6 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} → ({𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} ↔ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}))
20	19	biimpcd 248	. . . . 5 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → ({𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}))
21	14, 18, 20	e11 44364	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
22		df-sn 4634	. . . . . 6 ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
23	22	ax-gen 1790	. . . . 5 ⊢ ∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
24		csbeq2 3897	. . . . . 6 ⊢ (∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵} → ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵})
25	24	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵} → ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}))
26	1, 23, 25	e10 44370	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   )
27		eqeq2 2738	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} ↔ ⦋𝐴 / 𝑥⦌{𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}))
28	27	biimpd 228	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} → ⦋𝐴 / 𝑥⦌{𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}))
29	21, 26, 28	e11 44364	. . 3 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
30		df-sn 4634	. . 3 ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}
31		eqeq2 2738	. . . 4 ⊢ ({⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵} ↔ ⦋𝐴 / 𝑥⦌{𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}))
32	31	biimprcd 249	. . 3 ⊢ (⦋𝐴 / 𝑥⦌{𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → ({⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}))
33	29, 30, 32	e10 44370	. 2 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}   )
34	33	in1 44247	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1532   = wceq 1534   ∈ wcel 2099  {cab 2703  [wsbc 3776  ⦋csb 3892  {csn 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-nul 4326  df-sn 4634  df-vd1 44246

This theorem is referenced by: (None)