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Mirrors > Home > ILE Home > Th. List > prlem1 | GIF version |
Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) |
Ref | Expression |
---|---|
prlem1.1 | ⊢ (𝜑 → (𝜂 ↔ 𝜒)) |
prlem1.2 | ⊢ (𝜓 → ¬ 𝜃) |
Ref | Expression |
---|---|
prlem1 | ⊢ (𝜑 → (𝜓 → (((𝜓 ∧ 𝜒) ∨ (𝜃 ∧ 𝜏)) → 𝜂))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prlem1.1 | . . . . 5 ⊢ (𝜑 → (𝜂 ↔ 𝜒)) | |
2 | 1 | biimprd 158 | . . . 4 ⊢ (𝜑 → (𝜒 → 𝜂)) |
3 | 2 | adantld 278 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜂)) |
4 | prlem1.2 | . . . . 5 ⊢ (𝜓 → ¬ 𝜃) | |
5 | 4 | pm2.21d 619 | . . . 4 ⊢ (𝜓 → (𝜃 → 𝜂)) |
6 | 5 | adantrd 279 | . . 3 ⊢ (𝜓 → ((𝜃 ∧ 𝜏) → 𝜂)) |
7 | 3, 6 | jaao 719 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝜓 ∧ 𝜒) ∨ (𝜃 ∧ 𝜏)) → 𝜂)) |
8 | 7 | ex 115 | 1 ⊢ (𝜑 → (𝜓 → (((𝜓 ∧ 𝜒) ∨ (𝜃 ∧ 𝜏)) → 𝜂))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in2 615 ax-io 709 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: (None) |
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