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Mirrors > Home > NFE Home > Th. List > mpan9 | GIF version |
Description: Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
Ref | Expression |
---|---|
mpan9.1 | ⊢ (φ → ψ) |
mpan9.2 | ⊢ (χ → (ψ → θ)) |
Ref | Expression |
---|---|
mpan9 | ⊢ ((φ ∧ χ) → θ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpan9.1 | . . 3 ⊢ (φ → ψ) | |
2 | mpan9.2 | . . 3 ⊢ (χ → (ψ → θ)) | |
3 | 1, 2 | syl5 28 | . 2 ⊢ (χ → (φ → θ)) |
4 | 3 | impcom 419 | 1 ⊢ ((φ ∧ χ) → 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: sylan 457 vtocl2gf 2916 vtocl3gf 2917 vtoclegft 2926 sbcthdv 3061 nnsucelr 4428 nnadjoin 4520 sfintfin 4532 funiunfv 5467 isorel 5489 caovcld 5622 caovcomg 5624 caovassg 5626 caovdig 5632 caovdirg 5633 |
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