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Theorem pm5.1im 251
Description: Two propositions are equivalent if they are both true. Closed form of 2th 252. Equivalent to a biimp 203-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 ↔ (𝜑𝜓))). (Contributed by Wolf Lammen, 12-May-2013.)
Assertion
Ref Expression
pm5.1im (𝜑 → (𝜓 → (𝜑𝜓)))

Proof of Theorem pm5.1im
StepHypRef Expression
1 ax-1 6 . 2 (𝜓 → (𝜑𝜓))
2 ax-1 6 . 2 (𝜑 → (𝜓𝜑))
31, 2impbid21d 199 1 (𝜑 → (𝜓 → (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194
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 195
This theorem is referenced by:  2thd  253  pm5.501  354
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