Theorem wl-2xor 37418

Description: In the recursive scheme

"(n+1)-xor" ↔ if-(𝜑, ¬ "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 37417). (Contributed by Wolf Lammen, 11-May-2024.)

Assertion

Ref	Expression
wl-2xor	⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ (𝜑 ⊻ 𝜓))

Proof of Theorem wl-2xor

Step	Hyp	Ref	Expression
1		ifpdfbi 1070	. 2 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ if-(¬ 𝜑, 𝜓, ¬ 𝜓))
2		df-xor 1511	. . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3		nbbn 383	. . 3 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
4	2, 3	bitr4i 278	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ (¬ 𝜑 ↔ 𝜓))
5		ifpn 1073	. 2 ⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(¬ 𝜑, 𝜓, ¬ 𝜓))
6	1, 4, 5	3bitr4ri 304	1 ⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ (𝜑 ⊻ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 206  if-wif 1062   ⊻ wxo 1510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-xor 1511

This theorem is referenced by: (None)