Theorem ralbid2 2491

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

Hypotheses

Ref	Expression
ralbid2.nf	⊢ Ⅎ𝑥𝜑
ralbid2.1	⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)))

Assertion

Ref	Expression
ralbid2	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Proof of Theorem ralbid2

Step	Hyp	Ref	Expression
1		ralbid2.nf	. . 3 ⊢ Ⅎ𝑥𝜑
2		ralbid2.1	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)))
3	1, 2	albid 1625	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)))
4		df-ral 2470	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
5		df-ral 2470	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))
6	3, 4, 5	3bitr4g 223	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1361  Ⅎwnf 1470   ∈ wcel 2158  ∀wral 2465

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 1457  ax-gen 1459  ax-4 1520

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-ral 2470

This theorem is referenced by:  strcollnft  15007