Theorem mprg 1249

Description: Modus ponens combined with restricted generalization.

Hypotheses

Ref	Expression
mprg.1	⊢ (∀x ∈ A φ → ψ)
mprg.2	⊢ (x ∈ A → φ)

Assertion

Ref	Expression
mprg	⊢ ψ

Proof of Theorem mprg

Step	Hyp	Ref	Expression
1		mprg.2	. . 3 ⊢ (x ∈ A → φ)
2	1	rgen 1247	. 2 ⊢ ∀x ∈ A φ
3		mprg.1	. 2 ⊢ (∀x ∈ A φ → ψ)
4	2, 3	ax-mp 6	1 ⊢ ψ

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

This theorem is referenced by:  r19.22i 1273  reuxfr2 1579  iuneq2i 2008  iineq2i 2009  rankuni 3533  ranklon 3540  projlem17 5209  goeq 5706

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-gen 677

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

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