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Theorem albid 2210
Description: Formula-building rule for universal quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
Hypotheses
Ref Expression
albid.1 𝑥𝜑
albid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
albid (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Proof of Theorem albid
StepHypRef Expression
1 albid.1 . . 3 𝑥𝜑
21nf5ri 2183 . 2 (𝜑 → ∀𝑥𝜑)
3 albid.2 . 2 (𝜑 → (𝜓𝜒))
42, 3albidh 1861 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778
This theorem is referenced by:  nfbidf  2212  dral1vOLD  2361  dral2  2431  dral1  2432  sb4b  2468  sbal1  2521  sbal2  2522  ralbidaOLD  3258  raleqf  3336  intab  4982  fin23lem32  10369  axrepndlem1  10617  axrepndlem2  10618  axrepnd  10619  axunnd  10621  axpowndlem2  10623  axpowndlem4  10625  axregndlem2  10628  axinfndlem1  10630  axinfnd  10631  axacndlem4  10635  axacndlem5  10636  axacnd  10637  iota5f  35449  exrecfnlem  36989  wl-equsald  37137  wl-equsaldv  37138  wl-sbnf1  37153  wl-2sb6d  37156  wl-sbalnae  37160  wl-mo2df  37168  wl-eudf  37170  wl-ax11-lem6  37188  wl-ax11-lem8  37190  ax12eq  38543  ax12el  38544  ax12v2-o  38551  unielss  42788
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