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Mirrors > Home > ILE Home > Th. List > 3adant1r | 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 |
---|---|
3adant1r | ⊢ (((𝜑 ∧ 𝜏) ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3adant1l.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
2 | 1 | 3expb 1182 | . . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) |
3 | 2 | adantlr 468 | . 2 ⊢ (((𝜑 ∧ 𝜏) ∧ (𝜓 ∧ 𝜒)) → 𝜃) |
4 | 3 | 3impb 1177 | 1 ⊢ (((𝜑 ∧ 𝜏) ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 964 |
This theorem is referenced by: 3adant2r 1211 3adant3r 1213 tfr1onlembacc 6239 tfr1onlembfn 6241 tfr1onlemaccex 6245 tfr1onlemres 6246 tfrcllembfn 6254 tfrcllemaccex 6258 tfrcllemres 6259 tfrcldm 6260 tfrcl 6261 mulassnqg 7192 prarloc 7311 prmuloc 7374 addasssrg 7564 axaddass 7680 |
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