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Metamath Proof Explorer

Theorem eqeqan12dALT

Description: Alternate proof of eqeqan12d . This proof has one more step but one fewer essential step. (Contributed by NM, 9-Aug-1994) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	eqeqan12dALT.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	eqeqan12dALT.2	⊢ ( 𝜓 → 𝐶 = 𝐷 )
Assertion	eqeqan12dALT	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		eqeqan12dALT.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqeqan12dALT.2	⊢ ( 𝜓 → 𝐶 = 𝐷 )
3		eqeq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )
4	1 2 3	syl2an	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )