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Metamath Proof Explorer

Theorem mprg

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

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

Proof

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