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Mirrors > Home > ILE Home > Th. List > jca31 | GIF version |
Description: Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.) |
Ref | Expression |
---|---|
jca31.1 | ⊢ (𝜑 → 𝜓) |
jca31.2 | ⊢ (𝜑 → 𝜒) |
jca31.3 | ⊢ (𝜑 → 𝜃) |
Ref | Expression |
---|---|
jca31 | ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∧ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jca31.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | jca31.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | 1, 2 | jca 301 | . 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒)) |
4 | jca31.3 | . 2 ⊢ (𝜑 → 𝜃) | |
5 | 3, 4 | jca 301 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∧ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia3 107 |
This theorem is referenced by: 3jca 1124 syl21anc 1174 f1oiso2 5620 nnnq0lem1 7066 prmuloc 7186 prsrlem1 7349 apreap 8125 lemulge11 8388 elnnz 8821 supinfneg 9144 infsupneg 9145 leexp1a 10071 faclbnd6 10213 zfz1isolem1 10306 oddpwdclemdc 11490 |
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