Theorem csbopabw 5527

Description: Move substitution into a class abstraction. Version of csbopab 5526 with a sethood antecedent but depending on fewer axioms. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
csbopabw	⊢ (𝐴 ∈ 𝑉 → ⦋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)   𝑉(𝑥, 𝑦, 𝑧)

Proof of Theorem csbopabw

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3849	. . 3 ⊢ (𝑤 = 𝐴 → ⦋𝑤 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑})
2		dfsbcq2 3741	. . . 4 ⊢ (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	opabbidv 5170	. . 3 ⊢ (𝑤 = 𝐴 → {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
4	1, 3	eqeq12d 2776	. 2 ⊢ (𝑤 = 𝐴 → (⦋𝑤 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} ↔ ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}))
5		vex 3454	. . 3 ⊢ 𝑤 ∈ V
6		nfs1v 2193	. . . 4 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
7	6	nfopab 5173	. . 3 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
8		sbequ12 2286	. . . 4 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
9	8	opabbidv 5170	. . 3 ⊢ (𝑥 = 𝑤 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑})
10	5, 7, 9	csbief 3880	. 2 ⊢ ⦋𝑤 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
11	4, 10	vtoclg 3517	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})

This proof depends on syntax axioms:   → wi 4   = wceq 1570  [wsb 2099   ∈ wcel 2145  [wsbc 3738  ⦋csb 3846  {copab 5166

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-sbc 3739  df-csb 3847  df-opab 5167

This theorem is used by:  csbcnv  5860  csbcnvgALTOLD  5862