Theorem exbid 2216

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 2188	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		albid.2	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	exbidh 1870	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 205  ∃wex 1781  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1782  df-nf 1786

This theorem is referenced by:  nfbidf  2217  drex2  2441  rexbida  3269  rexeqfOLD  3351  opabbid  5213  zfrepclf  5294  dfid3  5577  oprabbid  7473  axrepndlem1  10586  axrepndlem2  10587  axrepnd  10588  axpowndlem2  10592  axpowndlem3  10593  axpowndlem4  10594  axregnd  10598  axinfndlem1  10599  axinfnd  10600  axacndlem4  10604  axacndlem5  10605  axacnd  10606  opabdm  31835  opabrn  31836  pm14.122b  43172  pm14.123b  43175