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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 679 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑)) | |
2 | ancom 262 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
3 | 2 | notbii 626 | . . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ ¬ (𝜓 ∧ 𝜑)) |
4 | 1, 3 | anbi12i 447 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ ((𝜓 ∨ 𝜑) ∧ ¬ (𝜓 ∧ 𝜑))) |
5 | df-xor 1307 | . 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) | |
6 | df-xor 1307 | . 2 ⊢ ((𝜓 ⊻ 𝜑) ↔ ((𝜓 ∨ 𝜑) ∧ ¬ (𝜓 ∧ 𝜑))) | |
7 | 4, 5, 6 | 3bitr4i 210 | 1 ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 102 ↔ wb 103 ∨ wo 661 ⊻ wxo 1306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 |
This theorem depends on definitions: df-bi 115 df-xor 1307 |
This theorem is referenced by: rpnegap 8766 |
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