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Mirrors > Home > ILE Home > Th. List > syl13anc | 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) |
syl13anc.5 | ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) → 𝜂) |
Ref | Expression |
---|---|
syl13anc | ⊢ (𝜑 → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylXanc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | sylXanc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | sylXanc.3 | . . 3 ⊢ (𝜑 → 𝜃) | |
4 | sylXanc.4 | . . 3 ⊢ (𝜑 → 𝜏) | |
5 | 2, 3, 4 | 3jca 1084 | . 2 ⊢ (𝜑 → (𝜒 ∧ 𝜃 ∧ 𝜏)) |
6 | syl13anc.5 | . 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) → 𝜂) | |
7 | 1, 5, 6 | syl2anc 391 | 1 ⊢ (𝜑 → 𝜂) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 885 |
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 887 |
This theorem is referenced by: syl23anc 1142 syl33anc 1150 caovassd 5660 caovcand 5663 caovordid 5667 caovordd 5669 caovdid 5676 caovdird 5679 swoer 6134 swoord1 6135 swoord2 6136 prarloclem3 6595 fzosubel3 9052 iseqsplit 9238 iseqcaopr 9242 |
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