Theorem ceqex 2391

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 1376	. . 3 |- (x = A -> E.x x = A)
2		isset 2296	. . 3 |- (A e. _V <-> E.x x = A)
3	1, 2	sylibr 217	. 2 |- (x = A -> A e. _V)
4		eqeq2 1893	. . . 4 |- (y = A -> (x = y <-> x = A))
5	4	anbi1d 679	. . . . . 6 |- (y = A -> ((x = y /\ ph) <-> (x = A /\ ph)))
6	5	exbidv 1657	. . . . 5 |- (y = A -> (E.x(x = y /\ ph) <-> E.x(x = A /\ ph)))
7	6	bibi2d 680	. . . 4 |- (y = A -> ((ph <-> E.x(x = y /\ ph)) <-> (ph <-> E.x(x = A /\ ph))))
8	4, 7	imbi12d 688	. . 3 |- (y = A -> ((x = y -> (ph <-> E.x(x = y /\ ph))) <-> (x = A -> (ph <-> E.x(x = A /\ ph)))))
9		19.8a 1376	. . . . 5 |- ((x = y /\ ph) -> E.x(x = y /\ ph))
10	9	ex 402	. . . 4 |- (x = y -> (ph -> E.x(x = y /\ ph)))
11		ax-4 1319	. . . . . 6 |- (A.x(x = y -> ph) -> (x = y -> ph))
12	11	com12 14	. . . . 5 |- (x = y -> (A.x(x = y -> ph) -> ph))
13		visset 2295	. . . . . 6 |- y e. _V
14	13	alexeq 2390	. . . . 5 |- (A.x(x = y -> ph) <-> E.x(x = y /\ ph))
15	12, 14	syl5ibr 224	. . . 4 |- (x = y -> (E.x(x = y /\ ph) -> ph))
16	10, 15	impbid 574	. . 3 |- (x = y -> (ph <-> E.x(x = y /\ ph)))
17	8, 16	vtoclg 2346	. 2 |- (A e. _V -> (x = A -> (ph <-> E.x(x = A /\ ph))))
18	3, 17	mpcom 60	1 |- (x = A -> (ph <-> E.x(x = A /\ ph)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292

This theorem is referenced by:  ceqsexg 2392  sbc6g 2472  copsexgOLD 3538

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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