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Mirrors > Home > ILE Home > Th. List > pm5.18im | GIF version |
Description: One direction of pm5.18dc 873, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) |
Ref | Expression |
---|---|
pm5.18im | ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.19 696 | . 2 ⊢ ¬ (𝜓 ↔ ¬ 𝜓) | |
2 | bibi1 239 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜓))) | |
3 | 2 | notbid 657 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜓 ↔ ¬ 𝜓))) |
4 | 1, 3 | mpbiri 167 | 1 ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: xornbi 1376 |
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