Theorem csbopab 5574

Description: Move substitution into a class abstraction. Version of csbopabgALT 5575 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 3924	. . . 4 ⊢ (𝑤 = 𝐴 → ⦋𝑤 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑})
2		dfsbcq2 3807	. . . . 5 ⊢ (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	opabbidv 5232	. . . 4 ⊢ (𝑤 = 𝐴 → {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
4	1, 3	eqeq12d 2756	. . 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 3492	. . . 4 ⊢ 𝑤 ∈ V
6		nfs1v 2157	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
7	6	nfopab 5235	. . . 4 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
8		sbequ12 2252	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
9	8	opabbidv 5232	. . . 4 ⊢ (𝑥 = 𝑤 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑})
10	5, 7, 9	csbief 3956	. . 3 ⊢ ⦋𝑤 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
11	4, 10	vtoclg 3566	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
12		csbprc 4432	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = ∅)
13		sbcex 3814	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
14	13	con3i 154	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
15	14	nexdv 1935	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑧[𝐴 / 𝑥]𝜑)
16	15	nexdv 1935	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑦∃𝑧[𝐴 / 𝑥]𝜑)
17		opabn0 5572	. . . . 5 ⊢ ({⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} ≠ ∅ ↔ ∃𝑦∃𝑧[𝐴 / 𝑥]𝜑)
18	17	necon1bbii 2996	. . . 4 ⊢ (¬ ∃𝑦∃𝑧[𝐴 / 𝑥]𝜑 ↔ {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} = ∅)
19	16, 18	sylib 218	. . 3 ⊢ (¬ 𝐴 ∈ V → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} = ∅)
20	12, 19	eqtr4d 2783	. 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 1537  ∃wex 1777  [wsb 2064   ∈ wcel 2108  Vcvv 3488  [wsbc 3804  ⦋csb 3921  ∅c0 4352  {copab 5228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229

This theorem is referenced by:  csbmpt12  5576  csbcnv  5908