Theorem sbccsb2g 3166

Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)

Assertion

Ref	Expression
sbccsb2g	⊢ (A ∈ V → ([̣A / x]̣φ ↔ A ∈ [A / x]{x ∣ φ}))

Proof of Theorem sbccsb2g

Step	Hyp	Ref	Expression
1		abid 2341	. . 3 ⊢ (x ∈ {x ∣ φ} ↔ φ)
2	1	sbcbii 3102	. 2 ⊢ ([̣A / x]̣x ∈ {x ∣ φ} ↔ [̣A / x]̣φ)
3		sbcel12g 3152	. . 3 ⊢ (A ∈ V → ([̣A / x]̣x ∈ {x ∣ φ} ↔ [A / x]x ∈ [A / x]{x ∣ φ}))
4		csbvarg 3164	. . . 4 ⊢ (A ∈ V → [A / x]x = A)
5	4	eleq1d 2419	. . 3 ⊢ (A ∈ V → ([A / x]x ∈ [A / x]{x ∣ φ} ↔ A ∈ [A / x]{x ∣ φ}))
6	3, 5	bitrd 244	. 2 ⊢ (A ∈ V → ([̣A / x]̣x ∈ {x ∣ φ} ↔ A ∈ [A / x]{x ∣ φ}))
7	2, 6	syl5bbr 250	1 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ A ∈ [A / x]{x ∣ φ}))

Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  {cab 2339  [̣wsbc 3047  [csb 3137

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  df-csb 3138

This theorem is referenced by: (None)