Theorem fneq12d

Description: Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011)

	Ref	Expression
Hypotheses	fneq12d.1	⊢ ( 𝜑 → 𝐹 = 𝐺 )
	fneq12d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	fneq12d	⊢ ( 𝜑 → ( 𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fneq12d.1	⊢ ( 𝜑 → 𝐹 = 𝐺 )
2		fneq12d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3	1	fneq1d	⊢ ( 𝜑 → ( 𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴 ) )
4	2	fneq2d	⊢ ( 𝜑 → ( 𝐺 Fn 𝐴 ↔ 𝐺 Fn 𝐵 ) )
5	3 4	bitrd	⊢ ( 𝜑 → ( 𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵 ) )