Theorem biralda 1213

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

Hypotheses

Ref	Expression
biralda.1	⊢ (φ → ∀xφ)
biralda.2	⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))

Assertion

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

Proof of Theorem biralda

Step	Hyp	Ref	Expression
1		biralda.1	. . 3 ⊢ (φ → ∀xφ)
2		biralda.2	. . . . 5 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
3	2	exp 291	. . . 4 ⊢ (φ → (x ∈ A → (ψ ↔ χ)))
4	3	pm5.74d 444	. . 3 ⊢ (φ → ((x ∈ A → ψ) ↔ (x ∈ A → χ)))
5	1, 4	biald 782	. 2 ⊢ (φ → (∀x(x ∈ A → ψ) ↔ ∀x(x ∈ A → χ)))
6		df-ral 1205	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
7		df-ral 1205	. 2 ⊢ (∀x ∈ A χ ↔ ∀x(x ∈ A → χ))
8	5, 6, 7	3bitr4g 428	1 ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ A χ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  biraldva 1215  birald 1217  bi2ralda 1232  r19.15 1292  iineq2 2007  mapxpen 3390  xpmapenlem5 3395

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

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

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