Theorem ralbida 2629

Description: Formula-building rule for restricted universal quantifier (deduction rule). (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 668	. . 3 ⊢ (φ → ((x ∈ A → ψ) ↔ (x ∈ A → χ)))
4	1, 3	albid 1772	. 2 ⊢ (φ → (∀x(x ∈ A → ψ) ↔ ∀x(x ∈ A → χ)))
5		df-ral 2620	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
6		df-ral 2620	. 2 ⊢ (∀x ∈ A χ ↔ ∀x(x ∈ A → χ))
7	4, 5, 6	3bitr4g 279	1 ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ A χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  ralbidva  2631  ralbid  2633  2ralbida  2654  ralbi  2751