Theorem sbhypf 1929

Description: Introduce an explicit substitution into an implicit substitution hypothesis.

Hypotheses

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

Assertion

Ref	Expression
sbhypf	|- (y = A -> ([y / x]ph <-> ps))

Distinct variable groups:   x,A   x,y

Proof of Theorem sbhypf

Step	Hyp	Ref	Expression
1		visset 1804	. . 3 |- y e. V
2		eqeq1 1473	. . 3 |- (x = y -> (x = A <-> y = A))
3	1, 2	ceqsexv 1826	. 2 |- (E.x(x = y /\ x = A) <-> y = A)
4		hbs1 1327	. . . 4 |- ([y / x]ph -> A.x[y / x]ph)
5		sbhypf.1	. . . 4 |- (ps -> A.xps)
6	4, 5	hbbi 1007	. . 3 |- (([y / x]ph <-> ps) -> A.x([y / x]ph <-> ps))
7		sbequ12 1177	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
8	7	bicomd 519	. . . 4 |- (x = y -> ([y / x]ph <-> ph))
9		sbhypf.2	. . . 4 |- (x = A -> (ph <-> ps))
10	8, 9	sylan9bb 538	. . 3 |- ((x = y /\ x = A) -> ([y / x]ph <-> ps))
11	6, 10	19.23ai 1060	. 2 |- (E.x(x = y /\ x = A) -> ([y / x]ph <-> ps))
12	3, 11	sylbir 201	1 |- (y = A -> ([y / x]ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953  E.wex 977  [wsbc 1166

This theorem is referenced by:  ac6sf 4732

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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