Theorem spimv 1783

Description: A version of spim 1716 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 1508	. 2 |- F/ x ps
2		spimv.1	. 2 |- ( x = y -> ( ph -> ps ) )
3	1, 2	spim 1716	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   A.wal 1329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  aev  1784  ax16i  1830  spv  1832  reu6  2868  el  4097