Theorem csbopab 5534

Description: Move substitution into a class abstraction. Version of csbopabw 5535 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 3854	. . . 4 ⊢ (𝑤 = 𝐴 → ⦋𝑤 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑})
2		dfsbcq2 3745	. . . . 5 ⊢ (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	opabbidv 5174	. . . 4 ⊢ (𝑤 = 𝐴 → {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
4	1, 3	eqeq12d 2779	. . 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 3458	. . . 4 ⊢ 𝑤 ∈ V
6		nfs1v 2196	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
7	6	nfopab 5177	. . . 4 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
8		sbequ12 2290	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
9	8	opabbidv 5174	. . . 4 ⊢ (𝑥 = 𝑤 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑})
10	5, 7, 9	csbief 3885	. . 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 4370	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = ∅)
13		sbcex 3752	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
14	13	con3i 155	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
15	14	nexdv 1971	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑧[𝐴 / 𝑥]𝜑)
16	15	nexdv 1971	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑦∃𝑧[𝐴 / 𝑥]𝜑)
17		opabn0 5532	. . . . 5 ⊢ ({⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} ≠ ∅ ↔ ∃𝑦∃𝑧[𝐴 / 𝑥]𝜑)
18	17	necon1bbii 3008	. . . 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 2801	. 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}

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ∃wex 1812  [wsb 2101   ∈ wcel 2147  Vcvv 3454  [wsbc 3742  ⦋csb 3851  ∅c0 4281  {copab 5170

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171

This theorem is used by:  csbmpt12  5536  csbcnvOLD  5872