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Mirrors > Home > ILE Home > Th. List > mpbir3an | GIF version |
Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 9-Jan-2015.) |
Ref | Expression |
---|---|
mpbir3an.1 | ⊢ 𝜓 |
mpbir3an.2 | ⊢ 𝜒 |
mpbir3an.3 | ⊢ 𝜃 |
mpbir3an.4 | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) |
Ref | Expression |
---|---|
mpbir3an | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpbir3an.1 | . . 3 ⊢ 𝜓 | |
2 | mpbir3an.2 | . . 3 ⊢ 𝜒 | |
3 | mpbir3an.3 | . . 3 ⊢ 𝜃 | |
4 | 1, 2, 3 | 3pm3.2i 1127 | . 2 ⊢ (𝜓 ∧ 𝜒 ∧ 𝜃) |
5 | mpbir3an.4 | . 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) | |
6 | 4, 5 | mpbir 145 | 1 ⊢ 𝜑 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∧ w3a 930 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 932 |
This theorem is referenced by: limon 4367 limom 4465 issmo 6115 xpider 6430 1eluzge0 9219 2eluzge1 9221 0elunit 9610 1elunit 9611 4fvwrd4 9758 fzo0to42pr 9838 resqrexlemga 10635 sincos1sgn 11269 sincos2sgn 11270 qnnen 11736 strleun 11830 |
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