![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > syl31anc | GIF version |
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
Ref | Expression |
---|---|
sylXanc.1 | ⊢ (φ → ψ) |
sylXanc.2 | ⊢ (φ → χ) |
sylXanc.3 | ⊢ (φ → θ) |
sylXanc.4 | ⊢ (φ → τ) |
syl31anc.5 | ⊢ (((ψ ∧ χ ∧ θ) ∧ τ) → η) |
Ref | Expression |
---|---|
syl31anc | ⊢ (φ → η) |
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 | 3jca 1083 | . 2 ⊢ (φ → (ψ ∧ χ ∧ θ)) |
5 | sylXanc.4 | . 2 ⊢ (φ → τ) | |
6 | syl31anc.5 | . 2 ⊢ (((ψ ∧ χ ∧ θ) ∧ τ) → η) | |
7 | 4, 5, 6 | syl2anc 391 | 1 ⊢ (φ → η) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 884 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 df-3an 886 |
This theorem is referenced by: syl32anc 1142 stoic4b 1319 enq0tr 6417 ltmul12a 7607 lt2msq1 7632 ledivp1 7650 lemul1ad 7686 lemul2ad 7687 lediv2ad 8419 difelfznle 8763 expubnd 8965 |
Copyright terms: Public domain | W3C validator |