Theorem equs4 1510

Description: Lemma used in proofs of substitution properties.

Assertion

Ref	Expression
equs4	|- (A.x(x = y -> ph) -> E.x(x = y /\ ph))

Proof of Theorem equs4

Step	Hyp	Ref	Expression
1		simpr 350	. . . . . . . 8 |- ((A.x(x = y -> ph) /\ x = y) -> x = y)
2		ax-4 1319	. . . . . . . . 9 |- (A.x(x = y -> ph) -> (x = y -> ph))
3	2	imp 377	. . . . . . . 8 |- ((A.x(x = y -> ph) /\ x = y) -> ph)
4	1, 3	jc 153	. . . . . . 7 |- ((A.x(x = y -> ph) /\ x = y) -> -. (x = y -> -. ph))
5		ax-4 1319	. . . . . . 7 |- (A.x(x = y -> -. ph) -> (x = y -> -. ph))
6	4, 5	nsyl 131	. . . . . 6 |- ((A.x(x = y -> ph) /\ x = y) -> -. A.x(x = y -> -. ph))
7	6	ex 402	. . . . 5 |- (A.x(x = y -> ph) -> (x = y -> -. A.x(x = y -> -. ph)))
8		hbn1 1362	. . . . 5 |- (-. A.x(x = y -> -. ph) -> A.x -. A.x(x = y -> -. ph))
9	7, 8	syl6 25	. . . 4 |- (A.x(x = y -> ph) -> (x = y -> A.x -. A.x(x = y -> -. ph)))
10	9	a5i 1335	. . 3 |- (A.x(x = y -> ph) -> A.x(x = y -> A.x -. A.x(x = y -> -. ph)))
11		ax-9o 1481	. . 3 |- (A.x(x = y -> A.x -. A.x(x = y -> -. ph)) -> -. A.x(x = y -> -. ph))
12	10, 11	syl 12	. 2 |- (A.x(x = y -> ph) -> -. A.x(x = y -> -. ph))
13		equs3 1509	. 2 |- (E.x(x = y /\ ph) <-> -. A.x(x = y -> -. ph))
14	12, 13	sylibr 217	1 |- (A.x(x = y -> ph) -> E.x(x = y /\ ph))

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

This theorem is referenced by:  sb2 1541  equs45f 1569

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481

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

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