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Mirrors > Home > NFE Home > Th. List > mpd3an3 | GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) |
Ref | Expression |
---|---|
mpd3an3.2 | ⊢ ((φ ∧ ψ) → χ) |
mpd3an3.3 | ⊢ ((φ ∧ ψ ∧ χ) → θ) |
Ref | Expression |
---|---|
mpd3an3 | ⊢ ((φ ∧ ψ) → θ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpd3an3.2 | . 2 ⊢ ((φ ∧ ψ) → χ) | |
2 | mpd3an3.3 | . . 3 ⊢ ((φ ∧ ψ ∧ χ) → θ) | |
3 | 2 | 3expa 1151 | . 2 ⊢ (((φ ∧ ψ) ∧ χ) → θ) |
4 | 1, 3 | mpdan 649 | 1 ⊢ ((φ ∧ ψ) → θ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∧ w3a 934 |
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 df-3an 936 |
This theorem is referenced by: nnsucelr 4428 cupvalg 5812 ovcross 5845 pmvalg 6010 enadj 6060 ovmuc 6130 ovce 6172 addlecncs 6209 leconnnc 6218 ncslesuc 6267 |
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