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Mirrors > Home > NFE Home > Th. List > mpanr12 | GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.) |
Ref | Expression |
---|---|
mpanr12.1 | ⊢ ψ |
mpanr12.2 | ⊢ χ |
mpanr12.3 | ⊢ ((φ ∧ (ψ ∧ χ)) → θ) |
Ref | Expression |
---|---|
mpanr12 | ⊢ (φ → θ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpanr12.2 | . 2 ⊢ χ | |
2 | mpanr12.1 | . . 3 ⊢ ψ | |
3 | mpanr12.3 | . . 3 ⊢ ((φ ∧ (ψ ∧ χ)) → θ) | |
4 | 2, 3 | mpanr1 664 | . 2 ⊢ ((φ ∧ χ) → θ) |
5 | 1, 4 | mpan2 652 | 1 ⊢ (φ → θ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 177 df-an 360 |
This theorem is referenced by: sbthlem1 6203 |
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