Theorem vsbcint 1438

Description: Change variable of an implicit substitution hypothesis, introducing an explicit substitution. (Contributed by Raph Levien, 10-Apr-04.)

Hypothesis

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

Assertion

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

Distinct variable group(s):   ps,x,y   x,A

Proof of Theorem vsbcint

Step	Hyp	Ref	Expression
1		visset 1350	. . 3 |- y e. V
2		cleq1 1107	. . 3 |- (x = y -> (x = A <-> y = A))
3	1, 2	ceqsexv 1371	. 2 |- (E.x(x = y /\ x = A) <-> y = A)
4		hbs1 986	. . . 4 |- ([y / x]ph -> A.x[y / x]ph)
5		ax-17 925	. . . 4 |- (ps -> A.xps)
6	4, 5	hbbi 705	. . 3 |- (([y / x]ph <-> ps) -> A.x([y / x]ph <-> ps))
7		sbequ12 865	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
8	7	bicomd 399	. . . 4 |- (x = y -> ([y / x]ph <-> ph))
9		vsbcint.1	. . . 4 |- (x = A -> (ph <-> ps))
10	8, 9	sylan9bb 418	. . 3 |- ((x = y /\ x = A) -> ([y / x]ph <-> ps))
11	6, 10	19.23ai 746	. 2 |- (E.x(x = y /\ x = A) -> ([y / x]ph <-> ps))
12	3, 11	sylbir 176	1 |- (y = A -> ([y / x]ph <-> ps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = weq 797  [wsb 852   = wceq 1091

This theorem is referenced by:  nn0ind 4612

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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