Theorem alexbii 1863

Description: Biconditional form of aleximi 1862. (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 232	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	aleximi 1862	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4	1	biimprd 251	. . 3 ⊢ (𝜑 → (𝜒 → 𝜓))
5	4	aleximi 1862	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜒 → ∃𝑥𝜓))
6	3, 5	impbid 215	1 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839

This theorem depends on definitions:  df-bi 210  df-ex 1810

This theorem is referenced by:  exbi  1877  exbidh  1897  exintrbi  1921  eleq2d  2849  rexeq  3319  rexss  4011  ttrclselem2  9691  bnj956  35165  bj-2exbi  37224  bj-axreprepsep  37712