Theorem exbid 2223

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 2193	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		albid.2	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	exbidh 1868	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 209  ∃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 1911  ax-6 1970  ax-7 2015  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786

This theorem is referenced by:  nfbidf  2224  drex2  2453  rexbida  3277  rexeqf  3351  opabbid  5095  zfrepclf  5162  dfid3  5427  oprabbid  7198  axrepndlem1  10003  axrepndlem2  10004  axrepnd  10005  axpowndlem2  10009  axpowndlem3  10010  axpowndlem4  10011  axregnd  10015  axinfndlem1  10016  axinfnd  10017  axacndlem4  10021  axacndlem5  10022  axacnd  10023  opabdm  30375  opabrn  30376  pm14.122b  41127  pm14.123b  41130