Theorem sbc8g 3054

Description: This is the closest we can get to df-sbc 3048 if we start from dfsbcq 3049 (see its comments) and dfsbcq2 3050. (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 3049	. 2 ⊢ (y = A → ([̣y / x]̣φ ↔ [̣A / x]̣φ))
2		eleq1 2413	. 2 ⊢ (y = A → (y ∈ {x ∣ φ} ↔ A ∈ {x ∣ φ}))
3		df-clab 2340	. . 3 ⊢ (y ∈ {x ∣ φ} ↔ [y / x]φ)
4		equid 1676	. . . 4 ⊢ y = y
5		dfsbcq2 3050	. . . 4 ⊢ (y = y → ([y / x]φ ↔ [̣y / x]̣φ))
6	4, 5	ax-mp 5	. . 3 ⊢ ([y / x]φ ↔ [̣y / x]̣φ)
7	3, 6	bitr2i 241	. 2 ⊢ ([̣y / x]̣φ ↔ y ∈ {x ∣ φ})
8	1, 2, 7	vtoclbg 2916	1 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ A ∈ {x ∣ φ}))

Syntax hints:   → wi 4   ↔ wb 176  [wsb 1648   ∈ wcel 1710  {cab 2339  [̣wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by: (None)