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	|- F/xps
sbciegf.2	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
sbciegf	|- (A e. V -> ( [.A / x ].ph <-> ps))

Distinct variable group:   x,A

Allowed substitution hints:   ph(x)   ps(x)   V(x)

Proof of Theorem sbciegf

Step	Hyp	Ref	Expression
1		sbciegf.1	. 2 |- F/xps
2		sbciegf.2	. . 3 |- (x = A -> (ph <-> ps))
3	2	ax-gen 1546	. 2 |- A.x(x = A -> (ph <-> ps))
4		sbciegft 3077	. 2 |- ((A e. V /\ F/xps /\ A.x(x = A -> (ph <-> ps))) -> ( [.A / x ].ph <-> ps))
5	1, 3, 4	mp3an23 1269	1 |- (A e. V -> ( [.A / x ].ph <-> ps))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710   [.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:  sbcieg  3079  opelopabf  4712