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Mirrors > Home > ILE Home > Th. List > mpbi2and | GIF version |
Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) |
Ref | Expression |
---|---|
mpbi2and.1 | ⊢ (𝜑 → 𝜓) |
mpbi2and.2 | ⊢ (𝜑 → 𝜒) |
mpbi2and.3 | ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝜃)) |
Ref | Expression |
---|---|
mpbi2and | ⊢ (𝜑 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpbi2and.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | mpbi2and.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | 1, 2 | jca 306 | . 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒)) |
4 | mpbi2and.3 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝜃)) | |
5 | 3, 4 | mpbid 147 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: supisoti 7002 remim 10840 resqrtcl 11009 divalgmod 11902 oddpwdclemxy 12139 divnumden 12166 numdensq 12172 prmdivdiv 12207 4sqlem7 12352 ismgmid2 12678 mnd1 12724 iscmnd 12915 topgele 13160 lmcn2 13413 |
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