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Mirrors > Home > ILE Home > Th. List > mprg | GIF version |
Description: Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.) |
Ref | Expression |
---|---|
mprg.1 | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝜓) |
mprg.2 | ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Ref | Expression |
---|---|
mprg | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mprg.2 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝜑) | |
2 | 1 | rgen 2485 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝜑 |
3 | mprg.1 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝜓) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∀wral 2416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-gen 1425 |
This theorem depends on definitions: df-bi 116 df-ral 2421 |
This theorem is referenced by: reximia 2527 rmoimia 2886 iuneq2i 3831 iineq2i 3832 dfiun2 3847 dfiin2 3848 dfiun3 4798 dfiin3 4799 cnviinm 5080 ixpintm 6619 sumeq2i 11133 prodeq2i 11331 bj-omtrans 13154 |
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