Theorem alexeq 2906

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 2217	. . . . 5 |- ( y = A -> ( x = y <-> x = A ) )
3	2	anbi1d 465	. . . 4 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
4	3	exbidv 1849	. . 3 |- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
5	2	imbi1d 231	. . . 4 |- ( y = A -> ( ( x = y -> ph ) <-> ( x = A -> ph ) ) )
6	5	albidv 1848	. . 3 |- ( y = A -> ( A. x ( x = y -> ph ) <-> A. x ( x = A -> ph ) ) )
7		sb56 1910	. . 3 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
8	1, 4, 6, 7	vtoclb 2835	. 2 |- ( E. x ( x = A /\ ph ) <-> A. x ( x = A -> ph ) )
9	8	bicomi 132	1 |- ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778

This theorem is referenced by:  ceqex  2907