Theorem spimv 1857

Description: A version of spim 1784 with a distinct variable requirement instead of a bound-variable hypothesis. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
spimv.1	|- ( x = y -> ( ph -> ps ) )

Assertion

Ref	Expression
spimv	|- ( A. x ph -> ps )

Distinct variable group:   ps, x

Allowed substitution hints:   ph( x, y)   ps( y)

Proof of Theorem spimv

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 |- F/ x ps
2		spimv.1	. 2 |- ( x = y -> ( ph -> ps ) )
3	1, 2	spim 1784	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   A.wal 1393

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  aev  1858  ax16i  1904  spv  1906  cbvalvw  1966  reu6  2992  el  4262