Theorem spv 1812

Description: Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)

Hypothesis

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

Assertion

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

Distinct variable group:   ps, x

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

Proof of Theorem spv

Step	Hyp	Ref	Expression
1		spv.1	. . 3 |- ( x = y -> ( ph <-> ps ) )
2	1	biimpd 143	. 2 |- ( x = y -> ( ph -> ps ) )
3	2	spimv 1763	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495

This theorem depends on definitions:  df-bi 116  df-nf 1418

This theorem is referenced by:  chvarv  1885  ru  2875  nalset  4016  tfisi  4459  tfr1onlemsucfn  6189  tfr1onlemsucaccv  6190  tfr1onlembxssdm  6192  tfr1onlembfn  6193  tfr1onlemres  6198  tfri1dALT  6200  tfrcllemsucfn  6202  tfrcllemsucaccv  6203  tfrcllembxssdm  6205  tfrcllembfn  6206  tfrcllemres  6211  findcard2  6734  findcard2s  6735  bj-nalset  12776