Theorem csbopab 5555

Description: Move substitution into a class abstraction. Version of csbopabgALT 5556 without a sethood antecedent but depending on more axioms. (Contributed by NM, 6-Aug-2007.) (Revised by NM, 23-Aug-2018.)

Assertion

Ref	Expression
csbopab	⊢ ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}

Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)

Proof of Theorem csbopab

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3896	. . . 4 ⊢ (𝑤 = 𝐴 → ⦋𝑤 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑})
2		dfsbcq2 3780	. . . . 5 ⊢ (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	opabbidv 5214	. . . 4 ⊢ (𝑤 = 𝐴 → {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
4	1, 3	eqeq12d 2748	. . 3 ⊢ (𝑤 = 𝐴 → (⦋𝑤 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} ↔ ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}))
5		vex 3478	. . . 4 ⊢ 𝑤 ∈ V
6		nfs1v 2153	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
7	6	nfopab 5217	. . . 4 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
8		sbequ12 2243	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
9	8	opabbidv 5214	. . . 4 ⊢ (𝑥 = 𝑤 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑})
10	5, 7, 9	csbief 3928	. . 3 ⊢ ⦋𝑤 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
11	4, 10	vtoclg 3556	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
12		csbprc 4406	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = ∅)
13		sbcex 3787	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
14	13	con3i 154	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
15	14	nexdv 1939	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑧[𝐴 / 𝑥]𝜑)
16	15	nexdv 1939	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑦∃𝑧[𝐴 / 𝑥]𝜑)
17		opabn0 5553	. . . . 5 ⊢ ({⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} ≠ ∅ ↔ ∃𝑦∃𝑧[𝐴 / 𝑥]𝜑)
18	17	necon1bbii 2990	. . . 4 ⊢ (¬ ∃𝑦∃𝑧[𝐴 / 𝑥]𝜑 ↔ {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} = ∅)
19	16, 18	sylib 217	. . 3 ⊢ (¬ 𝐴 ∈ V → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} = ∅)
20	12, 19	eqtr4d 2775	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
21	11, 20	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}

Syntax hints:  ¬ wn 3   = wceq 1541  ∃wex 1781  [wsb 2067   ∈ wcel 2106  Vcvv 3474  [wsbc 3777  ⦋csb 3893  ∅c0 4322  {copab 5210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211

This theorem is referenced by:  csbmpt12  5557  csbcnv  5883