Theorem sbhyp 1940

Description: Change variable of an implicit substitution hypothesis, introducing an explicit substitution. (Contributed by Raph Levien, 10-Apr-2004.)

Hypothesis

Ref	Expression
sbhyp.1	|- (x = A -> (ph <-> ps))

Assertion

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

Distinct variable groups:   ps,x   x,A   x,y

Proof of Theorem sbhyp

Step	Hyp	Ref	Expression
1		visset 1813	. . 3 |- y e. V
2		eqeq1 1481	. . 3 |- (x = y -> (x = A <-> y = A))
3	1, 2	ceqsexv 1835	. 2 |- (E.x(x = y /\ x = A) <-> y = A)
4		hbs1 1332	. . . 4 |- ([y / x]ph -> A.x[y / x]ph)
5		ax-17 971	. . . 4 |- (ps -> A.xps)
6	4, 5	hbbi 1010	. . 3 |- (([y / x]ph <-> ps) -> A.x([y / x]ph <-> ps))
7		sbequ12 1181	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
8	7	bicomd 521	. . . 4 |- (x = y -> ([y / x]ph <-> ph))
9		sbhyp.1	. . . 4 |- (x = A -> (ph <-> ps))
10	8, 9	sylan9bb 540	. . 3 |- ((x = y /\ x = A) -> ([y / x]ph <-> ps))
11	6, 10	19.23ai 1064	. 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   = wceq 956  E.wex 980  [wsbc 1170

This theorem is referenced by:  nn0ind-raph 6214

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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