Theorem csbopab 5540

Description: Move substitution into a class abstraction. Version of csbopabw 5541 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 3855	. . . 4 ⊢ (𝑤 = 𝐴 → ⦋𝑤 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑})
2		dfsbcq2 3746	. . . . 5 ⊢ (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	opabbidv 5176	. . . 4 ⊢ (𝑤 = 𝐴 → {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
4	1, 3	eqeq12d 2777	. . 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 3457	. . . 4 ⊢ 𝑤 ∈ V
6		nfs1v 2189	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
7	6	nfopab 5179	. . . 4 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
8		sbequ12 2285	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
9	8	opabbidv 5176	. . . 4 ⊢ (𝑥 = 𝑤 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑})
10	5, 7, 9	csbief 3886	. . 3 ⊢ ⦋𝑤 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
11	4, 10	vtoclg 3521	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
12		csbprc 4373	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = ∅)
13		sbcex 3753	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
14	13	con3i 155	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
15	14	nexdv 1964	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑧[𝐴 / 𝑥]𝜑)
16	15	nexdv 1964	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑦∃𝑧[𝐴 / 𝑥]𝜑)
17		opabn0 5538	. . . . 5 ⊢ ({⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} ≠ ∅ ↔ ∃𝑦∃𝑧[𝐴 / 𝑥]𝜑)
18	17	necon1bbii 3005	. . . 4 ⊢ (¬ ∃𝑦∃𝑧[𝐴 / 𝑥]𝜑 ↔ {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} = ∅)
19	16, 18	sylib 221	. . 3 ⊢ (¬ 𝐴 ∈ V → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} = ∅)
20	12, 19	eqtr4d 2799	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
21	11, 20	pm2.61i 184	1 ⊢ ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}

Syntax hints:  ¬ wn 3   = wceq 1568  ∃wex 1807  [wsb 2094   ∈ wcel 2141  Vcvv 3453  [wsbc 3743  ⦋csb 3852  ∅c0 4285  {copab 5172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-sn 4589  df-pr 4591  df-op 4595  df-opab 5173

This theorem is referenced by:  csbmpt12  5542  csbcnvOLD  5873