Theorem rexbid2 2459

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)

Hypotheses

Ref	Expression
rexbid2.nf	⊢ Ⅎ𝑥𝜑
rexbid2.1	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))

Assertion

Ref	Expression
rexbid2	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))

Proof of Theorem rexbid2

Step	Hyp	Ref	Expression
1		rexbid2.nf	. . 3 ⊢ Ⅎ𝑥𝜑
2		rexbid2.1	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
3	1, 2	exbid 1593	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)))
4		df-rex 2438	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
5		df-rex 2438	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
6	3, 4, 5	3bitr4g 222	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  Ⅎwnf 1437  ∃wex 1469   ∈ wcel 2125  ∃wrex 2433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-ial 1511

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-rex 2438

This theorem is referenced by: (None)