Theorem ralrexbid 3123

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 3103	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜃))
3		rexbi 3122	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜃) → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))
4	2, 3	syl 18	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wral 3080  ∃wrex 3090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3081  df-rex 3091

This theorem is used by:  r19.35  3124  r19.29  3129  r19.29r  3130  dmopab2rex  5906  fiun  7979  f1iun  7980  dmopab3rexdif  36337