Metamath Proof Explorer


Theorem wl-2xor

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 ) ↔ ( 𝜑𝜓 ) )

Proof

Step Hyp Ref Expression
1 ifpdfbi ( ( ¬ 𝜑𝜓 ) ↔ if- ( ¬ 𝜑 , 𝜓 , ¬ 𝜓 ) )
2 df-xor ( ( 𝜑𝜓 ) ↔ ¬ ( 𝜑𝜓 ) )
3 nbbn ( ( ¬ 𝜑𝜓 ) ↔ ¬ ( 𝜑𝜓 ) )
4 2 3 bitr4i ( ( 𝜑𝜓 ) ↔ ( ¬ 𝜑𝜓 ) )
5 ifpn ( if- ( 𝜑 , ¬ 𝜓 , 𝜓 ) ↔ if- ( ¬ 𝜑 , 𝜓 , ¬ 𝜓 ) )
6 1 4 5 3bitr4ri ( if- ( 𝜑 , ¬ 𝜓 , 𝜓 ) ↔ ( 𝜑𝜓 ) )