Theorem biraldv2 1221

Description: Formula-building rule for restricted universal quantifier (deduction rule).

Hypothesis

Ref	Expression
biraldv2.1	⊢ (φ → ((x ∈ A → ψ) ↔ (x ∈ B → χ)))

Assertion

Ref	Expression
biraldv2	⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B χ))

Distinct variable group(s):   φ,x

Proof of Theorem biraldv2

Step	Hyp	Ref	Expression
1		biraldv2.1	. . 3 ⊢ (φ → ((x ∈ A → ψ) ↔ (x ∈ B → χ)))
2	1	bialdv 935	. 2 ⊢ (φ → (∀x(x ∈ A → ψ) ↔ ∀x(x ∈ B → χ)))
3		df-ral 1205	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
4		df-ral 1205	. 2 ⊢ (∀x ∈ B χ ↔ ∀x(x ∈ B → χ))
5	2, 3, 4	3bitr4g 428	1 ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B χ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  oneqmini 2272  zornlem1 3603  iscard2 3660

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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