Theorem del40 839

Description: A distinctor elimination lemma. Formula-builder for existential quantifier.

Hypothesis

Ref	Expression
del40.1	|- (A.x x = y -> (ph -> ps))

Assertion

Ref	Expression
del40	|- (A.x x = y -> (E.xph -> E.yps))

Proof of Theorem del40

Step	Hyp	Ref	Expression
1		del40.1	. . . . 5 |- (A.x x = y -> (ph -> ps))
2	1	con3d 87	. . . 4 |- (A.x x = y -> (-. ps -> -. ph))
3	2	del35 836	. . 3 |- (A.x x = y -> (A.y -. ps -> A.x -. ph))
4	3	con3d 87	. 2 |- (A.x x = y -> (-. A.x -. ph -> -. A.y -. ps))
5		df-ex 679	. 2 |- (E.xph <-> -. A.x -. ph)
6		df-ex 679	. 2 |- (E.yps <-> -. A.y -. ps)
7	4, 5, 6	3imtr4g 426	1 |- (A.x x = y -> (E.xph -> E.yps))

Syntax hints:  -. wn 1   -> wi 2  A.wal 672  E.wex 678   = weq 797

This theorem is referenced by:  del43 856

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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