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  2441  rexbida  3250  rexeqfOLD  3333  opabbid  5175  zfrepclf  5249  dfid3  5539  oprabbid  7457  axrepndlem1  10552  axrepndlem2  10553  axrepnd  10554  axpowndlem2  10558  axpowndlem3  10559  axpowndlem4  10560  axregnd  10564  axinfndlem1  10565  axinfnd  10566  axacndlem4  10570  axacndlem5  10571  axacnd  10572  opabdm  32546  opabrn  32547  pm14.122b  44419  pm14.123b  44422  modelaxreplem3  44977