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Theorem exbid 2078
 Description: Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
Hypotheses
Ref Expression
albid.1 𝑥𝜑
albid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
exbid (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Proof of Theorem exbid
StepHypRef Expression
1 albid.1 . . 3 𝑥𝜑
21nf5ri 2053 . 2 (𝜑 → ∀𝑥𝜑)
3 albid.2 . 2 (𝜑 → (𝜓𝜒))
42, 3exbidh 1781 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701 This theorem is referenced by:  nfbidf  2079  mobid  2477  rexbida  3029  rexeqf  3112  opabbid  4647  zfrepclf  4705  dfid3  4954  oprabbid  6606  axrepndlem1  9293  axrepndlem2  9294  axrepnd  9295  axpowndlem2  9299  axpowndlem3  9300  axpowndlem4  9301  axregnd  9305  axinfndlem1  9306  axinfnd  9307  axacndlem4  9311  axacndlem5  9312  axacnd  9313  opabdm  28803  opabrn  28804  pm14.122b  37646  pm14.123b  37649
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