Theorem rexbi 2599

Description: Distribute a restricted existential quantifier over a biconditional. Theorem 19.18 of [Margaris] p. 90 with restricted quantification. (Contributed by Jim Kingdon, 21-Jan-2019.)

Assertion

Ref	Expression
rexbi	⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓))

Proof of Theorem rexbi

Step	Hyp	Ref	Expression
1		nfra1 2497	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)
2		rsp 2513	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)))
3	2	imp 123	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))
4	1, 3	rexbida 2461	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449  df-rex 2450

This theorem is referenced by:  rexrnmpo  5957