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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 1166 | . 2 ⊢ (𝜑 → (𝜒 ∧ 𝜃 ∧ 𝜏)) |
5 | mpbir3and.4 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) | |
6 | 4, 5 | mpbird 166 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 967 |
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 969 |
This theorem is referenced by: ixxss1 9834 ixxss2 9835 ixxss12 9836 ubioc1 9859 lbico1 9860 lbicc2 9914 ubicc2 9915 elicod 10194 modqelico 10263 zmodfz 10275 modqmuladdim 10296 addmodid 10301 phicl2 12140 isstruct2r 12399 lmtopcnp 12848 xmeter 13034 tgqioo 13145 suplociccreex 13200 dedekindicc 13209 ivthinclemlopn 13212 ivthinclemuopn 13214 sin0pilem2 13301 pilem3 13302 coseq0q4123 13353 |
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