Theorem rexbi 2446

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 2355	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)
2		rsp 2369	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)))
3	2	imp 115	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))
4	1, 3	rexbida 2321	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312

This theorem is referenced by:  rexrnmpt2  5616