Theorem ralbida 2464

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)

Hypotheses

Ref	Expression
ralbida.1	⊢ Ⅎ𝑥𝜑
ralbida.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem ralbida

Step	Hyp	Ref	Expression
1		ralbida.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		ralbida.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	pm5.74da 441	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜒)))
4	1, 3	albid 1608	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
5		df-ral 2453	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
6		df-ral 2453	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))
7	4, 5, 6	3bitr4g 222	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346  Ⅎwnf 1453   ∈ wcel 2141  ∀wral 2448

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 1440  ax-gen 1442  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453

This theorem is referenced by:  ralbidva  2466  ralbid  2468  2ralbida  2491  ralbi  2602  ismkvnex  7131  caucvgsrlemgt1  7757  iswomninnlem  14081