Theorem equs5e 1231

Description: A property related to substitution that unlike equs5 1254 doesn't require a distinctor antecedent.

Assertion

Ref	Expression
equs5e	|- (E.x(x = y /\ ph) -> A.x(x = y -> E.yph))

Proof of Theorem equs5e

Step	Hyp	Ref	Expression
1		equs3 1182	. 2 |- (E.x(x = y /\ ph) <-> -. A.x(x = y -> -. ph))
2		hbn1 1047	. . 3 |- (-. A.x(x = y -> -. ph) -> A.x -. A.x(x = y -> -. ph))
3		ax-11 999	. . . . . 6 |- (x = y -> (A.y -. ph -> A.x(x = y -> -. ph)))
4	3	con3d 95	. . . . 5 |- (x = y -> (-. A.x(x = y -> -. ph) -> -. A.y -. ph))
5	4	com12 11	. . . 4 |- (-. A.x(x = y -> -. ph) -> (x = y -> -. A.y -. ph))
6		df-ex 1013	. . . 4 |- (E.yph <-> -. A.y -. ph)
7	5, 6	syl6ibr 211	. . 3 |- (-. A.x(x = y -> -. ph) -> (x = y -> E.yph))
8	2, 7	19.21ai 1030	. 2 |- (-. A.x(x = y -> -. ph) -> A.x(x = y -> E.yph))
9	1, 8	sylbi 197	1 |- (E.x(x = y /\ ph) -> A.x(x = y -> E.yph))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 221  A.wal 986   = wceq 988  E.wex 1012

This theorem is referenced by:  sb4e 1236

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 995  ax-11 999  ax-4 1005  ax-5o 1007  ax-6o 1010

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1013

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