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Theorem pm5.1im 171
Description: Two propositions are equivalent if they are both true. Closed form of 2th 172. Equivalent to a bi1 116-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 5 . 2 (𝜓 → (𝜑𝜓))
2 ax-1 5 . 2 (𝜑 → (𝜓𝜑))
31, 2impbid21d 126 1 (𝜑 → (𝜓 → (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  2thd  173  pm5.501  242  dcextest  4396  snexxph  6659
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