Theorem r19.21ad 1261

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.)

Hypotheses

Ref	Expression
r19.21ad.1	⊢ (φ → ∀xφ)
r19.21ad.2	⊢ (ψ → ∀xψ)
r19.21ad.3	⊢ (φ → (ψ → (x ∈ A → χ)))

Assertion

Ref	Expression
r19.21ad	⊢ (φ → (ψ → ∀x ∈ A χ))

Proof of Theorem r19.21ad

Step	Hyp	Ref	Expression
1		r19.21ad.1	. . 3 ⊢ (φ → ∀xφ)
2		r19.21ad.2	. . 3 ⊢ (ψ → ∀xψ)
3		r19.21ad.3	. . 3 ⊢ (φ → (ψ → (x ∈ A → χ)))
4	1, 2, 3	19.21ad 741	. 2 ⊢ (φ → (ψ → ∀x(x ∈ A → χ)))
5		df-ral 1205	. 2 ⊢ (∀x ∈ A χ ↔ ∀x(x ∈ A → χ))
6	4, 5	syl6ibr 186	1 ⊢ (φ → (ψ → ∀x ∈ A χ))

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

This theorem is referenced by:  r19.21adv 1262  isotrALT 2936  tfrlem1 2949  mapxpen 3390

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

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

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