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Metamath Proof Explorer

Theorem pm5.1im

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

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

Proof

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