Theorem mprg 2564

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 2560	. 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 2177  ∀wral 2485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1473

This theorem depends on definitions:  df-bi 117  df-ral 2490

This theorem is referenced by:  reximia  2602  rmoimia  2976  iuneq2i  3947  iineq2i  3948  dfiun2  3963  dfiin2  3964  dfiun3  4942  dfiin3  4943  cnviinm  5229  ixpintm  6819  sumeq2i  11719  prodeq2i  11917  2sqlem1  15635  bj-omtrans  15966