Theorem functermclem 49112

Description: Lemma for functermc 49113. (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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐾𝑅𝐿)
3	1, 2	eqbrtrrd 5165	. . . 4 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐹𝑅𝐿)
4		functermclem.2	. . . . 5 ⊢ (𝜑 → (𝐹𝑅𝐿 ↔ 𝐿 = 𝐺))
5	4	biimpa 476	. . . 4 ⊢ ((𝜑 ∧ 𝐹𝑅𝐿) → 𝐿 = 𝐺)
6	3, 5	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐿 = 𝐺)
7	1, 6	jca 511	. 2 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → (𝐾 = 𝐹 ∧ 𝐿 = 𝐺))
8		simprl 771	. . 3 ⊢ ((𝜑 ∧ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)) → 𝐾 = 𝐹)
9	4	biimpar 477	. . . 4 ⊢ ((𝜑 ∧ 𝐿 = 𝐺) → 𝐹𝑅𝐿)
10	9	adantrl 716	. . 3 ⊢ ((𝜑 ∧ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)) → 𝐹𝑅𝐿)
11	8, 10	eqbrtrd 5163	. 2 ⊢ ((𝜑 ∧ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)) → 𝐾𝑅𝐿)
12	7, 11	impbida 801	1 ⊢ (𝜑 → (𝐾𝑅𝐿 ↔ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   class class class wbr 5141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142

This theorem is referenced by:  functermc  49113