Theorem sbc8g 2748

Description: This is the closest we can get to df-sbc 2742 if we start from dfsbcq 2743 (see its comments) and dfsbcq2 2744. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbc8g	⊢ (A ∈ 𝑉 → ([A / x]φ ↔ A ∈ {x ∣ φ}))

Proof of Theorem sbc8g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 2743	. 2 ⊢ (y = A → ([y / x]φ ↔ [A / x]φ))
2		eleq1 2082	. 2 ⊢ (y = A → (y ∈ {x ∣ φ} ↔ A ∈ {x ∣ φ}))
3		df-clab 2009	. . 3 ⊢ (y ∈ {x ∣ φ} ↔ [y / x]φ)
4		equid 1571	. . . 4 ⊢ y = y
5		dfsbcq2 2744	. . . 4 ⊢ (y = y → ([y / x]φ ↔ [y / x]φ))
6	4, 5	ax-mp 7	. . 3 ⊢ ([y / x]φ ↔ [y / x]φ)
7	3, 6	bitr2i 174	. 2 ⊢ ([y / x]φ ↔ y ∈ {x ∣ φ})
8	1, 2, 7	vtoclbg 2591	1 ⊢ (A ∈ 𝑉 → ([A / x]φ ↔ A ∈ {x ∣ φ}))

Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1374  [wsb 1627  {cab 2008  [wsbc 2741

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742

This theorem is referenced by:  bj-elssuniab  7037