Theorem sbcgf 3053

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Hypothesis

Ref	Expression
sbcgf.1	|- F/ x ph

Assertion

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

Proof of Theorem sbcgf

Step	Hyp	Ref	Expression
1		sbcgf.1	. 2 |- F/ x ph
2		sbctt 3052	. 2 |- ( ( A e. V /\ F/ x ph ) -> ( [. A / x ]. ph <-> ph ) )
3	1, 2	mpan2 425	1 |- ( A e. V -> ( [. A / x ]. ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1471   e. wcel 2164   [.wsbc 2985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986

This theorem is referenced by:  sbc19.21g  3054  sbcg  3055  sbcabel  3067