Description: In the recursive scheme
"(n+1)-xor" <-> if- ( ph , -. "n-xor" , "n-xor" )
we set n = 1 to formally arrive at an expression for "2-xor". It is based on "1-xor", that is known to be equivalent to its only input (see wl-1xor ). (Contributed by Wolf Lammen, 11-May-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | wl-2xor | ⊢ ( if- ( 𝜑 , ¬ 𝜓 , 𝜓 ) ↔ ( 𝜑 ⊻ 𝜓 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifpdfbi | ⊢ ( ( ¬ 𝜑 ↔ 𝜓 ) ↔ if- ( ¬ 𝜑 , 𝜓 , ¬ 𝜓 ) ) | |
2 | df-xor | ⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( 𝜑 ↔ 𝜓 ) ) | |
3 | nbbn | ⊢ ( ( ¬ 𝜑 ↔ 𝜓 ) ↔ ¬ ( 𝜑 ↔ 𝜓 ) ) | |
4 | 2 3 | bitr4i | ⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( ¬ 𝜑 ↔ 𝜓 ) ) |
5 | ifpn | ⊢ ( if- ( 𝜑 , ¬ 𝜓 , 𝜓 ) ↔ if- ( ¬ 𝜑 , 𝜓 , ¬ 𝜓 ) ) | |
6 | 1 4 5 | 3bitr4ri | ⊢ ( if- ( 𝜑 , ¬ 𝜓 , 𝜓 ) ↔ ( 𝜑 ⊻ 𝜓 ) ) |