Theorem speimfw 1645

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, 5-Aug-2017.)

Hypothesis

Ref	Expression
speimfw.2	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
speimfw	|- (-. A.x -. x = y -> (A.xph -> E.xps))

Proof of Theorem speimfw

Step	Hyp	Ref	Expression
1		speimfw.2	. . 3 |- (x = y -> (ph -> ps))
2	1	eximi 1576	. 2 |- (E.x x = y -> E.x(ph -> ps))
3		df-ex 1542	. 2 |- (E.x x = y <-> -. A.x -. x = y)
4		19.35 1600	. 2 |- (E.x(ph -> ps) <-> (A.xph -> E.xps))
5	2, 3, 4	3imtr3i 256	1 |- (-. A.x -. x = y -> (A.xph -> E.xps))

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:  spimfw  1646  19.2OLD  1700