Theorem ralrexbid 3241

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.)

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		df-ral 3066	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		ralrexbid.1	. . . . . 6 ⊢ (𝜑 → (𝜓 ↔ 𝜃))
3	2	imim2i 16	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜃)))
4	3	pm5.32d 580	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜃)))
5	4	alexbii 1840	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃)))
6	1, 5	sylbi 220	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃)))
7		df-rex 3067	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
8		df-rex 3067	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜃 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃))
9	6, 7, 8	3bitr4g 317	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541  ∃wex 1787   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-ral 3066  df-rex 3067

This theorem is referenced by:  dmopab2rex  5786  fiun  7716  f1iun  7717  dmopab3rexdif  33080