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Mirrors > Home > MPE Home > Th. List > 3adant1l | Structured version Visualization version GIF version |
Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) |
Ref | Expression |
---|---|
3adant1l.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
3adant1l | ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3adant1l.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
2 | 1 | 3expb 1258 | . . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) |
3 | 2 | adantll 746 | . 2 ⊢ (((𝜏 ∧ 𝜑) ∧ (𝜓 ∧ 𝜒)) → 𝜃) |
4 | 3 | 3impb 1252 | 1 ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 |
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 196 df-an 385 df-3an 1033 |
This theorem is referenced by: 3adant2l 1312 3adant3l 1314 cfsmolem 8975 axdc3lem4 9158 issubmnd 17141 maducoeval2 20265 cramerlem3 20314 restnlly 21095 efgh 24091 funvtxdm2val 25688 funiedgdm2val 25689 hasheuni 29474 matunitlindflem1 32575 pellex 36417 mendlmod 36782 disjf1o 38373 ssfiunibd 38464 mullimc 38683 mullimcf 38690 limclner 38718 sge0lefi 39291 isomenndlem 39420 hoicvr 39438 ovncvrrp 39454 |
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