Theorem speimfw 1876

Description: Specialization, with additional weakening (compared to 19.2 1892) 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-Dec-2017.)

Hypothesis

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

Assertion

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

Proof of Theorem speimfw

Step	Hyp	Ref	Expression
1		df-ex 1705	. . 3 |- ( E. x x = y <-> -. A. x -. x = y )
2	1	biimpri 218	. 2 |- ( -. A. x -. x = y -> E. x x = y )
3		speimfw.2	. . . 4 |- ( x = y -> ( ph -> ps ) )
4	3	com12 32	. . 3 |- ( ph -> ( x = y -> ps ) )
5	4	aleximi 1759	. 2 |- ( A. x ph -> ( E. x x = y -> E. x ps ) )
6	2, 5	syl5com 31	1 |- ( -. A. x -. x = y -> ( A. x ph -> E. x 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:  spimfw  1878