Theorem alexbii 1853

Description: Biconditional form of aleximi 1852. (Contributed by BJ, 16-Nov-2020.)

Hypothesis

Ref	Expression
alexbii.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
alexbii	⊢ (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Proof of Theorem alexbii

Step	Hyp	Ref	Expression
1		alexbii.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	biimpd 231	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	aleximi 1852	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4	1	biimprd 250	. . 3 ⊢ (𝜑 → (𝜒 → 𝜓))
5	4	aleximi 1852	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜒 → ∃𝑥𝜓))
6	3, 5	impbid 214	1 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1558  ∃wex 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829

This theorem depends on definitions:  df-bi 209  df-ex 1800

This theorem is referenced by:  exbi  1867  exbidh  1887  exintrbi  1911  eleq2d  2848  rexeq  3316  rexss  4010  ttrclselem2  9681  bnj956  35072  bj-2exbi  37074  bj-axreprepsep  37560