Theorem 19.21 1058

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "x is not free in ph."

Hypothesis

Ref	Expression
19.21.1	|- (ph -> A.xph)

Assertion

Ref	Expression
19.21	|- (A.x(ph -> ps) <-> (ph -> A.xps))

Proof of Theorem 19.21

Step	Hyp	Ref	Expression
1		19.20 996	. . 3 |- (A.x(ph -> ps) -> (A.xph -> A.xps))
2		19.21.1	. . 3 |- (ph -> A.xph)
3	1, 2	syl5 21	. 2 |- (A.x(ph -> ps) -> (ph -> A.xps))
4		hba1 1005	. . . 4 |- (A.xps -> A.xA.xps)
5	2, 4	hbim 1009	. . 3 |- ((ph -> A.xps) -> A.x(ph -> A.xps))
6		ax-4 975	. . . 4 |- (A.xps -> ps)
7	6	imim2i 17	. . 3 |- ((ph -> A.xps) -> (ph -> ps))
8	5, 7	19.21ai 1000	. 2 |- ((ph -> A.xps) -> A.x(ph -> ps))
9	3, 8	impbi 157	1 |- (A.x(ph -> ps) <-> (ph -> A.xps))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 956

This theorem is referenced by:  19.21-2 1059  stdpc5 1060  19.32 1088  hbim1 1105  19.21v 1287  cbvald 1322  ax15 1361  eu2 1398  moanim 1429

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977  ax-6o 980

This theorem depends on definitions:  df-bi 147

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