Theorem mprg 2684

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 2680	. 2 ⊢ ∀x ∈ A φ
3		mprg.1	. 2 ⊢ (∀x ∈ A φ → ψ)
4	2, 3	ax-mp 5	1 ⊢ ψ

Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546

This theorem depends on definitions:  df-bi 177  df-ral 2620

This theorem is referenced by:  reximia  2720  rmoimia  3037  iuneq2i  3988  iineq2i  3989  dfiun2  4002  dfiin2  4003  fnpw1fn  5854  dfnnc3  5886  enmap2lem2  6065  enmap1lem2  6071  enprmaplem2  6078  enprmaplem5  6081  fntcfn  6246  fnspac  6284  fnfreclem3  6320