Theorem ralrexbid 3255

Description: Formula-building rule for restricted existential quantifier, using a restricted universal quantifier to bind the quantified variable in the antecedent. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)

Hypothesis

Ref	Expression
ralrexbid.1	⊢ (𝜑 → (𝜓 ↔ 𝜃))

Assertion

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

Proof of Theorem ralrexbid

Step	Hyp	Ref	Expression
1		ralrexbid.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜃))
2	1	ralimi 3087	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜃))
3		rexbi 3173	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜃) → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))
4	2, 3	syl 17	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))

Syntax hints:   → wi 4   ↔ wb 205  ∀wral 3064  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-ral 3069  df-rex 3070

This theorem is referenced by:  dmopab2rex  5826  fiun  7785  f1iun  7786  dmopab3rexdif  33367