Theorem spimfw 1646

Description: Specialization, with additional weakening to allow bundling of x and y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)

Hypotheses

Ref	Expression
spimfw.1	|- (-. ps -> A.x -. ps)
spimfw.2	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
spimfw	|- (-. A.x -. x = y -> (A.xph -> ps))

Proof of Theorem spimfw

Step	Hyp	Ref	Expression
1		spimfw.2	. . 3 |- (x = y -> (ph -> ps))
2	1	speimfw 1645	. 2 |- (-. A.x -. x = y -> (A.xph -> E.xps))
3		df-ex 1542	. . 3 |- (E.xps <-> -. A.x -. ps)
4		spimfw.1	. . . 4 |- (-. ps -> A.x -. ps)
5	4	con1i 121	. . 3 |- (-. A.x -. ps -> ps)
6	3, 5	sylbi 187	. 2 |- (E.xps -> ps)
7	2, 6	syl6 29	1 |- (-. A.x -. x = y -> (A.xph -> ps))

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  spimw  1668