Theorem r19.21bi 1266

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

Hypothesis

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

Assertion

Ref	Expression
r19.21bi	⊢ ((φ ∧ x ∈ A) → ψ)

Proof of Theorem r19.21bi

Step	Hyp	Ref	Expression
1		r19.21bi.1	. . . 4 ⊢ (φ → ∀x ∈ A ψ)
2		df-ral 1205	. . . 4 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
3	1, 2	sylib 173	. . 3 ⊢ (φ → ∀x(x ∈ A → ψ))
4	3	19.21bi 742	. 2 ⊢ (φ → (x ∈ A → ψ))
5	4	imp 277	1 ⊢ ((φ ∧ x ∈ A) → ψ)

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

This theorem is referenced by:  rspec2 1267  rspec3 1268  r19.21be 1269  prcdpq 3891  prnmax 3893

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673

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

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