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Mirrors > Home > MPE Home > Th. List > bi2anan9r | Structured version Visualization version GIF version |
Description: Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.) |
Ref | Expression |
---|---|
bi2an9.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
bi2an9.2 | ⊢ (𝜃 → (𝜏 ↔ 𝜂)) |
Ref | Expression |
---|---|
bi2anan9r | ⊢ ((𝜃 ∧ 𝜑) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi2an9.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | bi2an9.2 | . . 3 ⊢ (𝜃 → (𝜏 ↔ 𝜂)) | |
3 | 1, 2 | bi2anan9 913 | . 2 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂))) |
4 | 3 | ancoms 468 | 1 ⊢ ((𝜃 ∧ 𝜑) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 df-an 385 |
This theorem is referenced by: efrn2lp 5020 ltsosr 9794 seqf1olem2 12703 seqf1o 12704 pcval 15387 usg2wlkeq 26236 fneval 31517 prtlem5 33162 rmydioph 36599 wepwsolem 36630 aomclem8 36649 uspgr2wlkeq 40854 |
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