Theorem exbid 2231

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 2203	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		albid.2	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	exbidh 1869	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 206  ∃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 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-ex 1782  df-nf 1786

This theorem is referenced by:  nfbidf  2232  drex2  2447  rexbida  3250  rexeqfOLD  3329  opabbid  5165  zfrepclf  5240  dfid3  5532  oprabbid  7435  axrepndlem1  10517  axrepndlem2  10518  axrepnd  10519  axpowndlem2  10523  axpowndlem3  10524  axpowndlem4  10525  axregnd  10529  axinfndlem1  10530  axinfnd  10531  axacndlem4  10535  axacndlem5  10536  axacnd  10537  opabdm  32707  opabrn  32708  pm14.122b  44808  pm14.123b  44811  modelaxreplem3  45365