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Metamath Proof Explorer

Theorem anbi1d

Description: Deduction adding a right conjunct to both sides of a logical equivalence. (Contributed by NM, 11-May-1993) (Proof shortened by Wolf Lammen, 16-Nov-2013)

		Ref	Expression
	Hypothesis	anbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	Assertion	anbi1d	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) ↔ ( 𝜒 ∧ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		anbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	a1d	⊢ ( 𝜑 → ( 𝜃 → ( 𝜓 ↔ 𝜒 ) ) )
3	2	pm5.32rd	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) ↔ ( 𝜒 ∧ 𝜃 ) ) )