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Mirrors > Home > ILE Home > Th. List > biadan2 | GIF version |
Description: Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) |
Ref | Expression |
---|---|
biadan2.1 | ⊢ (𝜑 → 𝜓) |
biadan2.2 | ⊢ (𝜓 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
biadan2 | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biadan2.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | 1 | pm4.71ri 384 | . 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑)) |
3 | biadan2.2 | . . 3 ⊢ (𝜓 → (𝜑 ↔ 𝜒)) | |
4 | 3 | pm5.32i 441 | . 2 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)) |
5 | 2, 4 | bitri 182 | 1 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: elab4g 2742 brab2a 4411 brab2ga 4433 elovmpt2 5721 eqop2 5824 elnnnn0 8331 elixx3g 8924 elfzo2 9160 1nprm 10496 |
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