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Metamath Proof Explorer

Theorem imbi1d

Description: Deduction adding a consequent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993) (Proof shortened by Wolf Lammen, 17-Sep-2013)

		Ref	Expression
	Hypothesis	imbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	Assertion	imbi1d	⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	biimprd	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
3	2	imim1d	⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) )
4	1	biimpd	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
5	4	imim1d	⊢ ( 𝜑 → ( ( 𝜒 → 𝜃 ) → ( 𝜓 → 𝜃 ) ) )
6	3 5	impbid	⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜃 ) ) )