Theorem r19.21v 1260

Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers.

Assertion

Ref	Expression
r19.21v	⊢ (∀x ∈ A (φ → ψ) ↔ (φ → ∀x ∈ A ψ))

Distinct variable group(s):   φ,x

Proof of Theorem r19.21v

Step	Hyp	Ref	Expression
1		bi2.04 141	. . . 4 ⊢ ((x ∈ A → (φ → ψ)) ↔ (φ → (x ∈ A → ψ)))
2	1	bial 695	. . 3 ⊢ (∀x(x ∈ A → (φ → ψ)) ↔ ∀x(φ → (x ∈ A → ψ)))
3		19.21v 942	. . 3 ⊢ (∀x(φ → (x ∈ A → ψ)) ↔ (φ → ∀x(x ∈ A → ψ)))
4	2, 3	bitr 151	. 2 ⊢ (∀x(x ∈ A → (φ → ψ)) ↔ (φ → ∀x(x ∈ A → ψ)))
5		df-ral 1205	. 2 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
6		df-ral 1205	. . 3 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
7	6	imbi2i 160	. 2 ⊢ ((φ → ∀x ∈ A ψ) ↔ (φ → ∀x(x ∈ A → ψ)))
8	4, 5, 7	3bitr4 158	1 ⊢ (∀x ∈ A (φ → ψ) ↔ (φ → ∀x ∈ A ψ))

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

This theorem is referenced by:  r19.32v 1297  ssiin 2024  dftr5 2044  tfinds2 2405  tfinds3 2406  tfr3 2964

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-6 675  ax-gen 677  ax-17 925

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

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