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Mirrors > Home > ILE Home > Th. List > mpanr1 | GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
Ref | Expression |
---|---|
mpanr1.1 | ⊢ 𝜓 |
mpanr1.2 | ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) |
Ref | Expression |
---|---|
mpanr1 | ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpanr1.1 | . 2 ⊢ 𝜓 | |
2 | mpanr1.2 | . . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) | |
3 | 2 | anassrs 398 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
4 | 1, 3 | mpanl2 432 | 1 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem is referenced by: mpanr12 436 axcnre 7713 rec11api 8537 divdiv23apzi 8549 recp1lt1 8681 divgt0i 8692 divge0i 8693 ltreci 8694 lereci 8695 lt2msqi 8696 le2msqi 8697 msq11i 8698 ltdiv23i 8708 ge0gtmnf 9636 sqrt11i 10936 sqrtmuli 10937 sqrtmsq2i 10939 sqrtlei 10940 sqrtlti 10941 cos01gt0 11505 |
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