Theorem alexeq 1888

Description: Two ways to express substitution of A for x in ph.

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

Proof of Theorem alexeq

Step	Hyp	Ref	Expression
1		alexeq.1	. . 3 |- A e. V
2		eqeq2 1487	. . . . 5 |- (y = A -> (x = y <-> x = A))
3	2	anbi1d 619	. . . 4 |- (y = A -> ((x = y /\ ph) <-> (x = A /\ ph)))
4	3	exbidv 1281	. . 3 |- (y = A -> (E.x(x = y /\ ph) <-> E.x(x = A /\ ph)))
5	2	imbi1d 615	. . . 4 |- (y = A -> ((x = y -> ph) <-> (x = A -> ph)))
6	5	albidv 1280	. . 3 |- (y = A -> (A.x(x = y -> ph) <-> A.x(x = A -> ph)))
7		sb56 1268	. . 3 |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))
8	1, 4, 6, 7	vtoclb 1848	. 2 |- (E.x(x = A /\ ph) <-> A.x(x = A -> ph))
9	8	bicomi 172	1 |- (A.x(x = A -> ph) <-> E.x(x = A /\ ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  Vcvv 1814

This theorem is referenced by:  ceqex 1889

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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