Theorem alexeq 1882

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 1482	. . . . 5 |- (y = A -> (x = y <-> x = A))
3	2	anbi1d 616	. . . 4 |- (y = A -> ((x = y /\ ph) <-> (x = A /\ ph)))
4	3	exbidv 1278	. . 3 |- (y = A -> (E.x(x = y /\ ph) <-> E.x(x = A /\ ph)))
5	2	imbi1d 612	. . . 4 |- (y = A -> ((x = y -> ph) <-> (x = A -> ph)))
6	5	albidv 1277	. . 3 |- (y = A -> (A.x(x = y -> ph) <-> A.x(x = A -> ph)))
7		sb56 1265	. . 3 |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))
8	1, 4, 6, 7	vtoclb 1842	. 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 953   = wceq 955   e. wcel 957  E.wex 979  Vcvv 1808

This theorem is referenced by:  ceqex 1883

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  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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