Theorem spv 1883

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 1834	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-nf 1484

This theorem is referenced by:  spvv  1931  cbvalvw  1943  chvarv  1965  ru  2997  nalset  4174  tfisi  4635  tfr1onlemsucfn  6426  tfr1onlemsucaccv  6427  tfr1onlembxssdm  6429  tfr1onlembfn  6430  tfr1onlemres  6435  tfri1dALT  6437  tfrcllemsucfn  6439  tfrcllemsucaccv  6440  tfrcllembxssdm  6442  tfrcllembfn  6443  tfrcllemres  6448  findcard2  6986  findcard2s  6987  bj-nalset  15831