Theorem sbbid 1244

Description: Deduction substituting both sides of a biconditional.

Hypotheses

Ref	Expression
sbbid.1	|- (ph -> A.xph)
sbbid.2	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
sbbid	|- (ph -> ([y / x]ps <-> [y / x]ch))

Proof of Theorem sbbid

Step	Hyp	Ref	Expression
1		sbbid.1	. . 3 |- (ph -> A.xph)
2		sbbid.2	. . 3 |- (ph -> (ps <-> ch))
3	1, 2	19.21ai 996	. 2 |- (ph -> A.x(ps <-> ch))
4		a4sbbi 1243	. 2 |- (A.x(ps <-> ch) -> ([y / x]ps <-> [y / x]ch))
5	3, 4	syl 10	1 |- (ph -> ([y / x]ps <-> [y / x]ch))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952  [wsbc 1168

This theorem is referenced by:  sbcom 1256  sbcom2 1332

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-10 964  ax-12 966  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170

Copyright terms: Public domain