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  2440  rexbida  3241  rexeqfOLD  3320  opabbid  5157  zfrepclf  5230  dfid3  5517  oprabbid  7414  axrepndlem1  10486  axrepndlem2  10487  axrepnd  10488  axpowndlem2  10492  axpowndlem3  10493  axpowndlem4  10494  axregnd  10498  axinfndlem1  10499  axinfnd  10500  axacndlem4  10504  axacndlem5  10505  axacnd  10506  opabdm  32556  opabrn  32557  pm14.122b  44406  pm14.123b  44409  modelaxreplem3  44964