Theorem equs5e 1567

Description: A property related to substitution that unlike equs5 1591 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 1509	. 2 |- (E.x(x = y /\ ph) <-> -. A.x(x = y -> -. ph))
2		hbn1 1362	. . 3 |- (-. A.x(x = y -> -. ph) -> A.x -. A.x(x = y -> -. ph))
3		ax-11 1309	. . . . . 6 |- (x = y -> (A.y -. ph -> A.x(x = y -> -. ph)))
4	3	con3d 111	. . . . 5 |- (x = y -> (-. A.x(x = y -> -. ph) -> -. A.y -. ph))
5	4	com12 14	. . . 4 |- (-. A.x(x = y -> -. ph) -> (x = y -> -. A.y -. ph))
6		df-ex 1327	. . . 4 |- (E.yph <-> -. A.y -. ph)
7	5, 6	syl6ibr 230	. . 3 |- (-. A.x(x = y -> -. ph) -> (x = y -> E.yph))
8	2, 7	19.21ai 1345	. 2 |- (-. A.x(x = y -> -. ph) -> A.x(x = y -> E.yph))
9	1, 8	sylbi 216	1 |- (E.x(x = y /\ ph) -> A.x(x = y -> E.yph))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326

This theorem is referenced by:  sb4e 1572

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-11 1309  ax-4 1319  ax-5o 1321  ax-6o 1324

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327

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