Theorem alexbii 1876

Description: Biconditional form of aleximi 1875. (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 221	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	aleximi 1875	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4	1	biimprd 240	. . 3 ⊢ (𝜑 → (𝜒 → 𝜓))
5	4	aleximi 1875	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜒 → ∃𝑥𝜓))
6	3, 5	impbid 204	1 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1599  ∃wex 1823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853

This theorem depends on definitions:  df-bi 199  df-ex 1824

This theorem is referenced by:  exbi  1891  exbidh  1912  exintrbi  1937  eleq2d  2845  bnj956  31446