Theorem functermclem 50285

Description: Lemma for functermc 50286. (Contributed by Zhi Wang, 17-Oct-2025.)

Hypotheses

Ref	Expression
functermclem.1	⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐾 = 𝐹)
functermclem.2	⊢ (𝜑 → (𝐹𝑅𝐿 ↔ 𝐿 = 𝐺))

Assertion

Ref	Expression
functermclem	⊢ (𝜑 → (𝐾𝑅𝐿 ↔ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)))

Proof of Theorem functermclem

Step	Hyp	Ref	Expression
1		functermclem.1	. . 3 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐾 = 𝐹)
2		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐾𝑅𝐿)
3	1, 2	eqbrtrrd 5135	. . . 4 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐹𝑅𝐿)
4		functermclem.2	. . . . 5 ⊢ (𝜑 → (𝐹𝑅𝐿 ↔ 𝐿 = 𝐺))
5	4	biimpa 481	. . . 4 ⊢ ((𝜑 ∧ 𝐹𝑅𝐿) → 𝐿 = 𝐺)
6	3, 5	syldan 602	. . 3 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐿 = 𝐺)
7	1, 6	jca 520	. 2 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → (𝐾 = 𝐹 ∧ 𝐿 = 𝐺))
8		simprl 782	. . 3 ⊢ ((𝜑 ∧ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)) → 𝐾 = 𝐹)
9	4	biimpar 482	. . . 4 ⊢ ((𝜑 ∧ 𝐿 = 𝐺) → 𝐹𝑅𝐿)
10	9	adantrl 728	. . 3 ⊢ ((𝜑 ∧ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)) → 𝐹𝑅𝐿)
11	8, 10	eqbrtrd 5133	. 2 ⊢ ((𝜑 ∧ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)) → 𝐾𝑅𝐿)
12	7, 11	impbida 812	1 ⊢ (𝜑 → (𝐾𝑅𝐿 ↔ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   class class class wbr 5109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110

This theorem is referenced by:  functermc  50286