Theorem sbc19.21g 3031

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 3004	. 2 |- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
2		sbcgf.1	. . . 4 |- F/ x ph
3	2	sbcgf 3030	. . 3 |- ( A e. V -> ( [. A / x ]. ph <-> ph ) )
4	3	imbi1d 231	. 2 |- ( A e. V -> ( ( [. A / x ]. ph -> [. A / x ]. ps ) <-> ( ph -> [. A / x ]. ps ) ) )
5	1, 4	bitrd 188	1 |- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( ph -> [. A / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1460   e. wcel 2148   [.wsbc 2962

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963

This theorem is referenced by: (None)