Theorem sbhypf 2904

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3171. (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 2862	. . 3 |- y e. _V
2		eqeq1 2359	. . 3 |- (x = y -> (x = A <-> y = A))
3	1, 2	ceqsexv 2894	. 2 |- (E.x(x = y /\ x = A) <-> y = A)
4		nfs1v 2106	. . . 4 |- F/x[y / x]ph
5		sbhypf.1	. . . 4 |- F/xps
6	4, 5	nfbi 1834	. . 3 |- F/x([y / x]ph <-> ps)
7		sbequ12 1919	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
8	7	bicomd 192	. . . 4 |- (x = y -> ([y / x]ph <-> ph))
9		sbhypf.2	. . . 4 |- (x = A -> (ph <-> ps))
10	8, 9	sylan9bb 680	. . 3 |- ((x = y /\ x = A) -> ([y / x]ph <-> ps))
11	6, 10	exlimi 1803	. 2 |- (E.x(x = y /\ x = A) -> ([y / x]ph <-> ps))
12	3, 11	sylbir 204	1 |- (y = A -> ([y / x]ph <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   = wceq 1642  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861

This theorem is referenced by:  mob2  3016  ralxpf  4827