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Mirrors > Home > ILE Home > Th. List > pm5.16 | GIF version |
Description: Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
pm5.16 | ⊢ ¬ ((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.19 706 | . 2 ⊢ ¬ (𝜓 ↔ ¬ 𝜓) | |
2 | simpl 109 | . . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) → (𝜑 ↔ 𝜓)) | |
3 | simpr 110 | . . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) → (𝜑 ↔ ¬ 𝜓)) | |
4 | 2, 3 | bitr3d 190 | . 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) → (𝜓 ↔ ¬ 𝜓)) |
5 | 1, 4 | mto 662 | 1 ⊢ ¬ ((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 |
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-in1 614 ax-in2 615 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: (None) |
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