Theorem sbcieg 3079

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)

Hypothesis

Ref	Expression
sbcieg.1	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
sbcieg	⊢ (A ∈ V → ([̣A / x]̣φ ↔ ψ))

Distinct variable groups:   x,A   ψ,x

Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem sbcieg

Step	Hyp	Ref	Expression
1		elex 2868	. 2 ⊢ (A ∈ V → A ∈ V)
2		nfv 1619	. . 3 ⊢ Ⅎxψ
3		sbcieg.1	. . 3 ⊢ (x = A → (φ ↔ ψ))
4	2, 3	sbciegf 3078	. 2 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ψ))
5	1, 4	syl 15	1 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2860  [̣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-3an 936  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:  sbcie  3081  ralsng  3766  rexsng  3767