Theorem ceqex 1883

Description: Equality implies equivalence with substitution.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem ceqex

Step	Hyp	Ref	Expression
1		19.8a 1028	. . 3 |- (x = A -> E.x x = A)
2		isset 1811	. . 3 |- (A e. V <-> E.x x = A)
3	1, 2	sylibr 200	. 2 |- (x = A -> A e. V)
4		eqeq2 1482	. . . 4 |- (y = A -> (x = y <-> x = A))
5	4	anbi1d 616	. . . . . 6 |- (y = A -> ((x = y /\ ph) <-> (x = A /\ ph)))
6	5	exbidv 1278	. . . . 5 |- (y = A -> (E.x(x = y /\ ph) <-> E.x(x = A /\ ph)))
7	6	bibi2d 617	. . . 4 |- (y = A -> ((ph <-> E.x(x = y /\ ph)) <-> (ph <-> E.x(x = A /\ ph))))
8	4, 7	imbi12d 625	. . 3 |- (y = A -> ((x = y -> (ph <-> E.x(x = y /\ ph))) <-> (x = A -> (ph <-> E.x(x = A /\ ph)))))
9		19.8a 1028	. . . . 5 |- ((x = y /\ ph) -> E.x(x = y /\ ph))
10	9	ex 373	. . . 4 |- (x = y -> (ph -> E.x(x = y /\ ph)))
11		ax-4 972	. . . . . 6 |- (A.x(x = y -> ph) -> (x = y -> ph))
12	11	com12 11	. . . . 5 |- (x = y -> (A.x(x = y -> ph) -> ph))
13		visset 1810	. . . . . 6 |- y e. V
14	13	alexeq 1882	. . . . 5 |- (A.x(x = y -> ph) <-> E.x(x = y /\ ph))
15	12, 14	syl5ibr 207	. . . 4 |- (x = y -> (E.x(x = y /\ ph) -> ph))
16	10, 15	impbid 515	. . 3 |- (x = y -> (ph <-> E.x(x = y /\ ph)))
17	8, 16	vtoclg 1844	. 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 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  E.wex 979  Vcvv 1808

This theorem is referenced by:  ceqsexg 1884  copsexg 2788

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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