Theorem pm5.1im 229

Description: Two propositions are equivalent if they are both true. Closed form of 2th 230. Equivalent to a bi1 178-like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version (φ ↔ (ψ ↔ (φ ↔ ψ))). (Contributed by Wolf Lammen, 12-May-2013.)

Assertion

Ref	Expression
pm5.1im	⊢ (φ → (ψ → (φ ↔ ψ)))

Proof of Theorem pm5.1im

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (ψ → (φ → ψ))
2		ax-1 6	. 2 ⊢ (φ → (ψ → φ))
3	1, 2	impbid21d 182	1 ⊢ (φ → (ψ → (φ ↔ ψ)))

Syntax hints:   → wi 4   ↔ wb 176

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 177

This theorem is referenced by:  2thd  231  pm5.501  330