Theorem mprg 2372

Description: Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)

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 2368	. 2 ⊢ ∀x ∈ A φ
3		mprg.1	. 2 ⊢ (∀x ∈ A φ → ψ)
4	2, 3	ax-mp 7	1 ⊢ ψ

Syntax hints:   → wi 4   ∈ wcel 1390  ∀wral 2300

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1335

This theorem depends on definitions:  df-bi 110  df-ral 2305

This theorem is referenced by:  reximia  2408  rmoimia  2735  iuneq2i  3666  iineq2i  3667  dfiun2  3682  dfiin2  3683  dfiun3  4534  dfiin3  4535  cnviinm  4802  bj-omtrans  9416