Theorem spimfw 1878

Description: Specialization, with additional weakening (compared to sp 2053) 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. x ph -> ps ) )

Proof of Theorem spimfw

Step	Hyp	Ref	Expression
1		spimfw.2	. . 3 |- ( x = y -> ( ph -> ps ) )
2	1	speimfw 1876	. 2 |- ( -. A. x -. x = y -> ( A. x ph -> E. x ps ) )
3		df-ex 1705	. . 3 |- ( E. x ps <-> -. A. x -. ps )
4		spimfw.1	. . . 4 |- ( -. ps -> A. x -. ps )
5	4	con1i 144	. . 3 |- ( -. A. x -. ps -> ps )
6	3, 5	sylbi 207	. 2 |- ( E. x ps -> ps )
7	2, 6	syl6 35	1 |- ( -. A. x -. x = y -> ( A. x ph -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  spimw  1926