Theorem rexbi 2640

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 2538	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)
2		rsp 2554	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)))
3	2	imp 124	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))
4	1, 3	rexbida 2502	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 105   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-ral 2490  df-rex 2491

This theorem is referenced by:  rexrnmpo  6071