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Mirrors > Home > MPE Home > Th. List > prlem2 | Structured version Visualization version GIF version |
Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 9-Dec-2012.) |
Ref | Expression |
---|---|
prlem2 | ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∧ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | simpl 482 | . . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜒) | |
3 | 1, 2 | orim12i 905 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) → (𝜑 ∨ 𝜒)) |
4 | 3 | pm4.71ri 560 | 1 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∧ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∨ wo 843 |
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 396 df-or 844 |
This theorem is referenced by: zfpair 5347 |
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