Theorem sbhypf 2597

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)

Hypotheses

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

Assertion

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

Distinct variable groups:   x,A   x,y

Allowed substitution hints:   ph(x,y)   ps(x,y)   A(y)

Proof of Theorem sbhypf

Step	Hyp	Ref	Expression
1		vex 2554	. . 3 |- y e. _V
2		eqeq1 2043	. . 3 |- (x = y -> (x = A <-> y = A))
3	1, 2	ceqsexv 2587	. 2 |- (E.x(x = y /\ x = A) <-> y = A)
4		nfs1v 1812	. . . 4 |- F/x[y / x]ph
5		sbhypf.1	. . . 4 |- F/xps
6	4, 5	nfbi 1478	. . 3 |- F/x([y / x]ph <-> ps)
7		sbequ12 1651	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
8	7	bicomd 129	. . . 4 |- (x = y -> ([y / x]ph <-> ph))
9		sbhypf.2	. . . 4 |- (x = A -> (ph <-> ps))
10	8, 9	sylan9bb 435	. . 3 |- ((x = y /\ x = A) -> ([y / x]ph <-> ps))
11	6, 10	exlimi 1482	. 2 |- (E.x(x = y /\ x = A) -> ([y / x]ph <-> ps))
12	3, 11	sylbir 125	1 |- (y = A -> ([y / x]ph <-> ps))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   F/wnf 1346  E.wex 1378  [wsb 1642

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  mob2  2715  tfisi  4253  ralxpf  4425  rexxpf  4426  nn0ind-raph  8131