Theorem mprg 2683

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 2679	. 2 ⊢ ∀x ∈ A φ
3		mprg.1	. 2 ⊢ (∀x ∈ A φ → ψ)
4	2, 3	ax-mp 8	1 ⊢ ψ

Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2614

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

This theorem depends on definitions:  df-bi 177  df-ral 2619

This theorem is referenced by:  reximia  2719  rmoimia  3036  iuneq2i  3987  iineq2i  3988  dfiun2  4001  dfiin2  4002  fnpw1fn  5853  dfnnc3  5885  enmap2lem2  6064  enmap1lem2  6070  enprmaplem2  6077  enprmaplem5  6080  fntcfn  6245  fnspac  6283  fnfreclem3  6319