Theorem r19.20sdv 1257

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90.

Hypothesis

Ref	Expression
r19.20sdv.1	⊢ (φ → (ψ → χ))

Assertion

Ref	Expression
r19.20sdv	⊢ (φ → (∀x ∈ A ψ → ∀x ∈ A χ))

Distinct variable group(s):   φ,x

Proof of Theorem r19.20sdv

Step	Hyp	Ref	Expression
1		r19.20sdv.1	. . 3 ⊢ (φ → (ψ → χ))
2	1	adantr 306	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ → χ))
3	2	r19.20dva 1256	1 ⊢ (φ → (∀x ∈ A ψ → ∀x ∈ A χ))

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

This theorem is referenced by:  tfindsg 2402  abianfp 3000  rankval3 3525  bndrank 3526  cfub 3703

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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