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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 392 | . 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑)) |
3 | biadan2.2 | . . 3 ⊢ (𝜓 → (𝜑 ↔ 𝜒)) | |
4 | 3 | pm5.32i 454 | . 2 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)) |
5 | 2, 4 | bitri 184 | 1 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
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-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: elab4g 2886 elpwb 3584 ssdifsn 3719 brab2a 4676 brab2ga 4698 elovmpo 6066 eqop2 6173 elnnnn0 9208 elixx3g 9888 elfzo2 10136 1nprm 12097 |
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