Theorem ralbidv2 2637

Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Apr-1997.)

Hypothesis

Ref	Expression
ralbidv2.1	⊢ (φ → ((x ∈ A → ψ) ↔ (x ∈ B → χ)))

Assertion

Ref	Expression
ralbidv2	⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B χ))

Distinct variable group:   φ,x

Allowed substitution hints:   ψ(x)   χ(x)   A(x)   B(x)

Proof of Theorem ralbidv2

Step	Hyp	Ref	Expression
1		ralbidv2.1	. . 3 ⊢ (φ → ((x ∈ A → ψ) ↔ (x ∈ B → χ)))
2	1	albidv 1625	. 2 ⊢ (φ → (∀x(x ∈ A → ψ) ↔ ∀x(x ∈ B → χ)))
3		df-ral 2620	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
4		df-ral 2620	. 2 ⊢ (∀x ∈ B χ ↔ ∀x(x ∈ B → χ))
5	2, 3, 4	3bitr4g 279	1 ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B χ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  ∀wral 2615

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616

This proof depends on definitions:  df-bi 177  df-ral 2620

This theorem is used by:  ralss  3333