Theorem sbciegf 3078

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
sbciegf.1	⊢ Ⅎxψ
sbciegf.2	⊢ (x = A → (φ ↔ ψ))

Assertion

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

Distinct variable group:   x,A

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

Proof of Theorem sbciegf

Step	Hyp	Ref	Expression
1		sbciegf.1	. 2 ⊢ Ⅎxψ
2		sbciegf.2	. . 3 ⊢ (x = A → (φ ↔ ψ))
3	2	ax-gen 1546	. 2 ⊢ ∀x(x = A → (φ ↔ ψ))
4		sbciegft 3077	. 2 ⊢ ((A ∈ V ∧ Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ))) → ([̣A / x]̣φ ↔ ψ))
5	1, 3, 4	mp3an23 1269	1 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  [̣wsbc 3047

This proof depends on 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 proof 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 used by:  sbcieg  3079  opelopabf  4712