Theorem r19.21ai 1258

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

Hypotheses

Ref	Expression
r19.21ai.1	⊢ (φ → ∀xφ)
r19.21ai.2	⊢ (φ → (x ∈ A → ψ))

Assertion

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

Proof of Theorem r19.21ai

Step	Hyp	Ref	Expression
1		r19.21ai.1	. . 3 ⊢ (φ → ∀xφ)
2		r19.21ai.2	. . 3 ⊢ (φ → (x ∈ A → ψ))
3	1, 2	19.21ai 740	. 2 ⊢ (φ → ∀x(x ∈ A → ψ))
4		df-ral 1205	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
5	3, 4	sylibr 175	1 ⊢ (φ → ∀x ∈ A ψ)

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

This theorem is referenced by:  r19.21aiv 1259  r19.22d 1276  r19.12 1281  zfrep6 2744  fnopabg 2745  isotrALT 2936  tfr3 2964  mapxpen 3390  aceq6b 3565  ac6lem 3575

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-ral 1205

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