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Mirrors > Home > NFE Home > Th. List > syl12anc | GIF version |
Description: Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.) |
Ref | Expression |
---|---|
sylXanc.1 | ⊢ (φ → ψ) |
sylXanc.2 | ⊢ (φ → χ) |
sylXanc.3 | ⊢ (φ → θ) |
syl12anc.4 | ⊢ ((ψ ∧ (χ ∧ θ)) → τ) |
Ref | Expression |
---|---|
syl12anc | ⊢ (φ → τ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylXanc.1 | . . 3 ⊢ (φ → ψ) | |
2 | sylXanc.2 | . . 3 ⊢ (φ → χ) | |
3 | sylXanc.3 | . . 3 ⊢ (φ → θ) | |
4 | 1, 2, 3 | jca32 521 | . 2 ⊢ (φ → (ψ ∧ (χ ∧ θ))) |
5 | syl12anc.4 | . 2 ⊢ ((ψ ∧ (χ ∧ θ)) → τ) | |
6 | 4, 5 | syl 15 | 1 ⊢ (φ → τ) |
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: syl22anc 1183 raaan 3657 raaanv 3658 nndisjeq 4429 prepeano4 4451 ssfin 4470 ncfinraise 4481 ncfinlower 4483 nnpweq 4523 peano4 4557 f1oiso2 5500 frecsuc 6322 |
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