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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 646 | . . 3 ⊢ ((φ ∨ ψ) ↔ (ψ ∨ φ)) | |
2 | ancom 253 | . . . 4 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ)) | |
3 | 2 | notbii 593 | . . 3 ⊢ (¬ (φ ∧ ψ) ↔ ¬ (ψ ∧ φ)) |
4 | 1, 3 | anbi12i 433 | . 2 ⊢ (((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)) ↔ ((ψ ∨ φ) ∧ ¬ (ψ ∧ φ))) |
5 | df-xor 1266 | . 2 ⊢ ((φ ⊻ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ))) | |
6 | df-xor 1266 | . 2 ⊢ ((ψ ⊻ φ) ↔ ((ψ ∨ φ) ∧ ¬ (ψ ∧ φ))) | |
7 | 4, 5, 6 | 3bitr4i 201 | 1 ⊢ ((φ ⊻ ψ) ↔ (ψ ⊻ φ)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ↔ wb 98 ∨ wo 628 ⊻ wxo 1265 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 |
This theorem depends on definitions: df-bi 110 df-xor 1266 |
This theorem is referenced by: rpnegap 8390 |
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