Theorem ceqex 2970

Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)

Assertion

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

Distinct variable group:   x,A

Allowed substitution hint:   ph(x)

Proof of Theorem ceqex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.8a 1756	. . 3 |- (x = A -> E.x x = A)
2		isset 2864	. . 3 |- (A e. _V <-> E.x x = A)
3	1, 2	sylibr 203	. 2 |- (x = A -> A e. _V)
4		eqeq2 2362	. . . 4 |- (y = A -> (x = y <-> x = A))
5	4	anbi1d 685	. . . . . 6 |- (y = A -> ((x = y /\ ph) <-> (x = A /\ ph)))
6	5	exbidv 1626	. . . . 5 |- (y = A -> (E.x(x = y /\ ph) <-> E.x(x = A /\ ph)))
7	6	bibi2d 309	. . . 4 |- (y = A -> ((ph <-> E.x(x = y /\ ph)) <-> (ph <-> E.x(x = A /\ ph))))
8	4, 7	imbi12d 311	. . 3 |- (y = A -> ((x = y -> (ph <-> E.x(x = y /\ ph))) <-> (x = A -> (ph <-> E.x(x = A /\ ph)))))
9		19.8a 1756	. . . . 5 |- ((x = y /\ ph) -> E.x(x = y /\ ph))
10	9	ex 423	. . . 4 |- (x = y -> (ph -> E.x(x = y /\ ph)))
11		vex 2863	. . . . . 6 |- y e. _V
12	11	alexeq 2969	. . . . 5 |- (A.x(x = y -> ph) <-> E.x(x = y /\ ph))
13		sp 1747	. . . . . 6 |- (A.x(x = y -> ph) -> (x = y -> ph))
14	13	com12 27	. . . . 5 |- (x = y -> (A.x(x = y -> ph) -> ph))
15	12, 14	syl5bir 209	. . . 4 |- (x = y -> (E.x(x = y /\ ph) -> ph))
16	10, 15	impbid 183	. . 3 |- (x = y -> (ph <-> E.x(x = y /\ ph)))
17	8, 16	vtoclg 2915	. 2 |- (A e. _V -> (x = A -> (ph <-> E.x(x = A /\ ph))))
18	3, 17	mpcom 32	1 |- (x = A -> (ph <-> E.x(x = A /\ ph)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  ceqsexg  2971  sbc6g  3072