Theorem sbieh 1814

Description: Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1815 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)

Hypotheses

Ref	Expression
sbieh.1	|- ( ps -> A. x ps )
sbieh.2	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Proof of Theorem sbieh

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( ph -> ph )
2	1	hbth 1487	. . 3 |- ( ( ph -> ph ) -> A. x ( ph -> ph ) )
3		sbieh.1	. . . 4 |- ( ps -> A. x ps )
4	3	a1i 9	. . 3 |- ( ( ph -> ph ) -> ( ps -> A. x ps ) )
5		sbieh.2	. . . 4 |- ( x = y -> ( ph <-> ps ) )
6	5	a1i 9	. . 3 |- ( ( ph -> ph ) -> ( x = y -> ( ph <-> ps ) ) )
7	2, 4, 6	sbiedh 1811	. 2 |- ( ( ph -> ph ) -> ( [ y / x ] ph <-> ps ) )
8	1, 7	ax-mp 5	1 |- ( [ y / x ] ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   [wsb 1786

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-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-sb 1787

This theorem is referenced by:  sbie  1815  sbco2vlem  1973  equsb3lem  1979  sbco2yz  1992  dvelimf  2044  elsb1  2185  elsb2  2186