Theorem alexeq 2969

Description: Two ways to express substitution of A for x in ph. (Contributed by NM, 2-Mar-1995.)

Hypothesis

Ref	Expression
alexeq.1	|- A e. _V

Assertion

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

Distinct variable group:   x,A

Allowed substitution hint:   ph(x)

Proof of Theorem alexeq

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alexeq.1	. . 3 |- A e. _V
2		eqeq2 2362	. . . . 5 |- (y = A -> (x = y <-> x = A))
3	2	anbi1d 685	. . . 4 |- (y = A -> ((x = y /\ ph) <-> (x = A /\ ph)))
4	3	exbidv 1626	. . 3 |- (y = A -> (E.x(x = y /\ ph) <-> E.x(x = A /\ ph)))
5	2	imbi1d 308	. . . 4 |- (y = A -> ((x = y -> ph) <-> (x = A -> ph)))
6	5	albidv 1625	. . 3 |- (y = A -> (A.x(x = y -> ph) <-> A.x(x = A -> ph)))
7		sb56 2098	. . 3 |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))
8	1, 4, 6, 7	vtoclb 2913	. 2 |- (E.x(x = A /\ ph) <-> A.x(x = A -> ph))
9	8	bicomi 193	1 |- (A.x(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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  ceqex  2970