Theorem sbieh 1673

Description: Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1674 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 1352	. . 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 1670	. 2 |- ( ( ph -> ph ) -> ( [ y / x ] ph <-> ps ) )
8	1, 7	ax-mp 7	1 |- ( [ y / x ] ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  sbie  1674  sbco2vlem  1820  equsb3lem  1824  sbco2yz  1837  elsb3  1852  elsb4  1853  dvelimf  1891