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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 635 | . 2 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂))) |
4 | 3 | ancoms 457 | 1 ⊢ ((𝜃 ∧ 𝜑) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 |
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 206 df-an 395 |
This theorem is referenced by: efrn2lp 5657 ltsosr 11091 seqf1olem2 14012 seqf1o 14013 pcval 16781 sltlpss 27638 uspgr2wlkeq 29170 satf0op 34666 fmlafvel 34674 fneval 35540 prtlem5 38033 prjspval 41647 rmydioph 42055 wepwsolem 42086 aomclem8 42105 sprsymrelfolem2 46459 |
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