Theorem pm5.1im 172

Description: Two propositions are equivalent if they are both true. Closed form of 2th 173. Equivalent to a biimp 117-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 127	1 ⊢ (𝜑 → (𝜓 → (𝜑 ↔ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  2thd  174  pm5.501  243  dcextest  4558  snexxph  6915