Theorem spv 1884

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 1835	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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485

This theorem is referenced by:  spvv  1932  cbvalvw  1944  chvarv  1966  ru  3004  nalset  4190  tfisi  4653  tfr1onlemsucfn  6449  tfr1onlemsucaccv  6450  tfr1onlembxssdm  6452  tfr1onlembfn  6453  tfr1onlemres  6458  tfri1dALT  6460  tfrcllemsucfn  6462  tfrcllemsucaccv  6463  tfrcllembxssdm  6465  tfrcllembfn  6466  tfrcllemres  6471  findcard2  7012  findcard2s  7013  bj-nalset  16030