Theorem functermclem

Description: Lemma for functermc . (Contributed by Zhi Wang, 17-Oct-2025)

	Ref	Expression
Hypotheses	functermclem.1	⊢ ( ( 𝜑 ∧ 𝐾 𝑅 𝐿 ) → 𝐾 = 𝐹 )
	functermclem.2	⊢ ( 𝜑 → ( 𝐹 𝑅 𝐿 ↔ 𝐿 = 𝐺 ) )
Assertion	functermclem	⊢ ( 𝜑 → ( 𝐾 𝑅 𝐿 ↔ ( 𝐾 = 𝐹 ∧ 𝐿 = 𝐺 ) ) )

Proof

Step	Hyp	Ref	Expression
1		functermclem.1	⊢ ( ( 𝜑 ∧ 𝐾 𝑅 𝐿 ) → 𝐾 = 𝐹 )
2		functermclem.2	⊢ ( 𝜑 → ( 𝐹 𝑅 𝐿 ↔ 𝐿 = 𝐺 ) )
3		simpr	⊢ ( ( 𝜑 ∧ 𝐾 𝑅 𝐿 ) → 𝐾 𝑅 𝐿 )
4	1 3	eqbrtrrd	⊢ ( ( 𝜑 ∧ 𝐾 𝑅 𝐿 ) → 𝐹 𝑅 𝐿 )
5	2	biimpa	⊢ ( ( 𝜑 ∧ 𝐹 𝑅 𝐿 ) → 𝐿 = 𝐺 )
6	4 5	syldan	⊢ ( ( 𝜑 ∧ 𝐾 𝑅 𝐿 ) → 𝐿 = 𝐺 )
7	1 6	jca	⊢ ( ( 𝜑 ∧ 𝐾 𝑅 𝐿 ) → ( 𝐾 = 𝐹 ∧ 𝐿 = 𝐺 ) )
8		simprl	⊢ ( ( 𝜑 ∧ ( 𝐾 = 𝐹 ∧ 𝐿 = 𝐺 ) ) → 𝐾 = 𝐹 )
9	2	biimpar	⊢ ( ( 𝜑 ∧ 𝐿 = 𝐺 ) → 𝐹 𝑅 𝐿 )
10	9	adantrl	⊢ ( ( 𝜑 ∧ ( 𝐾 = 𝐹 ∧ 𝐿 = 𝐺 ) ) → 𝐹 𝑅 𝐿 )
11	8 10	eqbrtrd	⊢ ( ( 𝜑 ∧ ( 𝐾 = 𝐹 ∧ 𝐿 = 𝐺 ) ) → 𝐾 𝑅 𝐿 )
12	7 11	impbida	⊢ ( 𝜑 → ( 𝐾 𝑅 𝐿 ↔ ( 𝐾 = 𝐹 ∧ 𝐿 = 𝐺 ) ) )