Theorem ceqex 1410

Description: Equality implies equivalence with substitution.

Assertion

Ref	Expression
ceqex	|- (x = A -> (ph <-> E.x(x = A /\ ph)))

Distinct variable group(s):   x,A

Proof of Theorem ceqex

Step	Hyp	Ref	Expression
1		19.8a 712	. . 3 |- (x = A -> E.x x = A)
2		isset 1351	. . 3 |- (A e. V <-> E.x x = A)
3	1, 2	sylibr 175	. 2 |- (x = A -> A e. V)
4		cleq2 1110	. . . 4 |- (y = A -> (x = y <-> x = A))
5	4	anbi1d 469	. . . . . 6 |- (y = A -> ((x = y /\ ph) <-> (x = A /\ ph)))
6	5	biexdv 936	. . . . 5 |- (y = A -> (E.x(x = y /\ ph) <-> E.x(x = A /\ ph)))
7	6	bibi2d 470	. . . 4 |- (y = A -> ((ph <-> E.x(x = y /\ ph)) <-> (ph <-> E.x(x = A /\ ph))))
8	4, 7	imbi12d 474	. . 3 |- (y = A -> ((x = y -> (ph <-> E.x(x = y /\ ph))) <-> (x = A -> (ph <-> E.x(x = A /\ ph)))))
9		19.8a 712	. . . . 5 |- ((x = y /\ ph) -> E.x(x = y /\ ph))
10	9	exp 291	. . . 4 |- (x = y -> (ph -> E.x(x = y /\ ph)))
11		ax-4 673	. . . . . 6 |- (A.x(x = y -> ph) -> (x = y -> ph))
12	11	com12 13	. . . . 5 |- (x = y -> (A.x(x = y -> ph) -> ph))
13		visset 1350	. . . . . 6 |- y e. V
14	13	alexeq 1409	. . . . 5 |- (A.x(x = y -> ph) <-> E.x(x = y /\ ph))
15	12, 14	syl5ibr 182	. . . 4 |- (x = y -> (E.x(x = y /\ ph) -> ph))
16	10, 15	impbid 397	. . 3 |- (x = y -> (ph <-> E.x(x = y /\ ph)))
17	8, 16	vtoclg 1383	. 2 |- (A e. V -> (x = A -> (ph <-> E.x(x = A /\ ph))))
18	3, 17	mpcom 49	1 |- (x = A -> (ph <-> E.x(x = A /\ ph)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  ceqsexg 1411  copsexg 1902

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-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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