Theorem alexeq 2890

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 2206	. . . . 5 |- ( y = A -> ( x = y <-> x = A ) )
3	2	anbi1d 465	. . . 4 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
4	3	exbidv 1839	. . 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 1838	. . 3 |- ( y = A -> ( A. x ( x = y -> ph ) <-> A. x ( x = A -> ph ) ) )
7		sb56 1900	. . 3 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
8	1, 4, 6, 7	vtoclb 2821	. 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 1362   = wceq 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by:  ceqex  2891