Theorem alexeq 2670

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 2049	. . . . 5 |- ( y = A -> ( x = y <-> x = A ) )
3	2	anbi1d 438	. . . 4 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
4	3	exbidv 1706	. . 3 |- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
5	2	imbi1d 220	. . . 4 |- ( y = A -> ( ( x = y -> ph ) <-> ( x = A -> ph ) ) )
6	5	albidv 1705	. . 3 |- ( y = A -> ( A. x ( x = y -> ph ) <-> A. x ( x = A -> ph ) ) )
7		sb56 1765	. . 3 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
8	1, 4, 6, 7	vtoclb 2611	. 2 |- ( E. x ( x = A /\ ph ) <-> A. x ( x = A -> ph ) )
9	8	bicomi 123	1 |- ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  ceqex  2671