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Mirrors > Home > ILE Home > Th. List > xorcom | GIF version |
Description: ⊻ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.) |
Ref | Expression |
---|---|
xorcom | ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orcom 728 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑)) | |
2 | ancom 266 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
3 | 2 | notbii 668 | . . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ ¬ (𝜓 ∧ 𝜑)) |
4 | 1, 3 | anbi12i 460 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ ((𝜓 ∨ 𝜑) ∧ ¬ (𝜓 ∧ 𝜑))) |
5 | df-xor 1376 | . 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) | |
6 | df-xor 1376 | . 2 ⊢ ((𝜓 ⊻ 𝜑) ↔ ((𝜓 ∨ 𝜑) ∧ ¬ (𝜓 ∧ 𝜑))) | |
7 | 4, 5, 6 | 3bitr4i 212 | 1 ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 ∨ wo 708 ⊻ wxo 1375 |
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 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-xor 1376 |
This theorem is referenced by: rpnegap 9682 |
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