Theorem sbc19.21g 2977

Description: Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)

Hypothesis

Ref	Expression
sbcgf.1	|- F/ x ph

Assertion

Ref	Expression
sbc19.21g	|- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( ph -> [. A / x ]. ps ) ) )

Proof of Theorem sbc19.21g

Step	Hyp	Ref	Expression
1		sbcimg 2950	. 2 |- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
2		sbcgf.1	. . . 4 |- F/ x ph
3	2	sbcgf 2976	. . 3 |- ( A e. V -> ( [. A / x ]. ph <-> ph ) )
4	3	imbi1d 230	. 2 |- ( A e. V -> ( ( [. A / x ]. ph -> [. A / x ]. ps ) <-> ( ph -> [. A / x ]. ps ) ) )
5	1, 4	bitrd 187	1 |- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( ph -> [. A / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1436   e. wcel 1480   [.wsbc 2909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910

This theorem is referenced by: (None)