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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 385 | . 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑)) |
3 | biadan2.2 | . . 3 ⊢ (𝜓 → (𝜑 ↔ 𝜒)) | |
4 | 3 | pm5.32i 443 | . 2 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)) |
5 | 2, 4 | bitri 183 | 1 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 |
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 |
This theorem is referenced by: elab4g 2765 elpwb 3443 ssdifsn 3574 brab2a 4504 brab2ga 4526 elovmpt2 5859 eqop2 5962 elnnnn0 8777 elixx3g 9380 elfzo2 9622 1nprm 11435 |
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