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Metamath Proof Explorer

Theorem pm4.38

Description: Theorem *4.38 of WhiteheadRussell p. 118. (Contributed by NM, 3-Jan-2005)

		Ref	Expression
	Assertion	pm4.38	⊢ ( ( ( 𝜑 ↔ 𝜒 ) ∧ ( 𝜓 ↔ 𝜃 ) ) → ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜃 ) ) )

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( ( 𝜑 ↔ 𝜒 ) ∧ ( 𝜓 ↔ 𝜃 ) ) → ( 𝜑 ↔ 𝜒 ) )
2		simpr	⊢ ( ( ( 𝜑 ↔ 𝜒 ) ∧ ( 𝜓 ↔ 𝜃 ) ) → ( 𝜓 ↔ 𝜃 ) )
3	1 2	anbi12d	⊢ ( ( ( 𝜑 ↔ 𝜒 ) ∧ ( 𝜓 ↔ 𝜃 ) ) → ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜃 ) ) )