Theorem albid 1821

Description: Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

Ref	Expression
albid.1	|- F/ x ph
albid.2	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
albid	|- ( ph -> ( A. x ps <-> A. x ch ) )

Proof of Theorem albid

Step	Hyp	Ref	Expression
1		albid.1	. . 3 |- F/ x ph
2	1	nfri 1810	. 2 |- ( ph -> A. x ph )
3		albid.2	. 2 |- ( ph -> ( ps <-> ch ) )
4	2, 3	albidh 1643	1 |- ( ph -> ( A. x ps <-> A. x ch ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1368   F/wnf 1590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794

This theorem depends on definitions:  df-bi 185  df-ex 1588  df-nf 1591

This theorem is referenced by:  nfbidf  1823  dral2  2023  dral1  2024  ax12v2  2040  sbal1  2178  sbal2  2180  sbal2OLD  2191  ax12eq  2249  ax12el  2250  ax12v2-o  2257  eubid  2281  ralbidaOLD  2834  raleqf  3011  intab  4258  fin23lem32  8616  axrepndlem1  8859  axrepndlem2  8860  axrepnd  8861  axunnd  8863  axpowndlem2  8865  axpowndlem2OLD  8866  axpowndlem4  8869  axregndlem2  8872  axinfndlem1  8875  axinfnd  8876  axacndlem4  8880  axacndlem5  8881  axacnd  8882  funcnvmptOLD  26122  iota5f  27517  wl-equsald  28508  wl-sbnf1  28519  wl-2sb6d  28524  wl-sbalnae  28528  wl-mo2dnae  28535  wl-ax11-lem6  28546  wl-ax11-lem8  28548  bj-dral1v  32563  bj-axc15v  32567