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Mirrors > Home > ILE Home > Th. List > mpbir3and | GIF version |
Description: Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.) |
Ref | Expression |
---|---|
mpbir3and.1 | ⊢ (𝜑 → 𝜒) |
mpbir3and.2 | ⊢ (𝜑 → 𝜃) |
mpbir3and.3 | ⊢ (𝜑 → 𝜏) |
mpbir3and.4 | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) |
Ref | Expression |
---|---|
mpbir3and | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpbir3and.1 | . . 3 ⊢ (𝜑 → 𝜒) | |
2 | mpbir3and.2 | . . 3 ⊢ (𝜑 → 𝜃) | |
3 | mpbir3and.3 | . . 3 ⊢ (𝜑 → 𝜏) | |
4 | 1, 2, 3 | 3jca 1084 | . 2 ⊢ (𝜑 → (𝜒 ∧ 𝜃 ∧ 𝜏)) |
5 | mpbir3and.4 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) | |
6 | 4, 5 | mpbird 156 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∧ 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: ixxss1 8773 ixxss2 8774 ixxss12 8775 ubioc1 8798 lbico1 8799 lbicc2 8852 ubicc2 8853 modqelico 9176 |
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