Theorem exbid 2224

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

Hypotheses

Ref	Expression
albid.1	⊢ Ⅎ𝑥𝜑
albid.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
exbid	⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Proof of Theorem exbid

Step	Hyp	Ref	Expression
1		albid.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	nf5ri 2196	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		albid.2	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	exbidh 1867	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784

This theorem is referenced by:  nfbidf  2225  drex2  2447  rexbida  3258  rexeqfOLD  3341  opabbid  5189  zfrepclf  5266  dfid3  5556  oprabbid  7477  axrepndlem1  10611  axrepndlem2  10612  axrepnd  10613  axpowndlem2  10617  axpowndlem3  10618  axpowndlem4  10619  axregnd  10623  axinfndlem1  10624  axinfnd  10625  axacndlem4  10629  axacndlem5  10630  axacnd  10631  opabdm  32596  opabrn  32597  pm14.122b  44414  pm14.123b  44417  modelaxreplem3  44972