Theorem spv 1874

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 144	. 2 |- ( x = y -> ( ph -> ps ) )
3	2	spimv 1825	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362

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-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  spvv  1922  cbvalvw  1934  chvarv  1956  ru  2988  nalset  4163  tfisi  4623  tfr1onlemsucfn  6398  tfr1onlemsucaccv  6399  tfr1onlembxssdm  6401  tfr1onlembfn  6402  tfr1onlemres  6407  tfri1dALT  6409  tfrcllemsucfn  6411  tfrcllemsucaccv  6412  tfrcllembxssdm  6414  tfrcllembfn  6415  tfrcllemres  6420  findcard2  6950  findcard2s  6951  bj-nalset  15541