Theorem functermclem 49997

Description: Lemma for functermc 49998. (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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐾𝑅𝐿)
3	1, 2	eqbrtrrd 5096	. . . 4 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐹𝑅𝐿)
4		functermclem.2	. . . . 5 ⊢ (𝜑 → (𝐹𝑅𝐿 ↔ 𝐿 = 𝐺))
5	4	biimpa 477	. . . 4 ⊢ ((𝜑 ∧ 𝐹𝑅𝐿) → 𝐿 = 𝐺)
6	3, 5	syldan 597	. . 3 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐿 = 𝐺)
7	1, 6	jca 516	. 2 ⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → (𝐾 = 𝐹 ∧ 𝐿 = 𝐺))
8		simprl 776	. . 3 ⊢ ((𝜑 ∧ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)) → 𝐾 = 𝐹)
9	4	biimpar 478	. . . 4 ⊢ ((𝜑 ∧ 𝐿 = 𝐺) → 𝐹𝑅𝐿)
10	9	adantrl 722	. . 3 ⊢ ((𝜑 ∧ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)) → 𝐹𝑅𝐿)
11	8, 10	eqbrtrd 5094	. 2 ⊢ ((𝜑 ∧ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)) → 𝐾𝑅𝐿)
12	7, 11	impbida 806	1 ⊢ (𝜑 → (𝐾𝑅𝐿 ↔ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   class class class wbr 5072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073

This theorem is referenced by:  functermc  49998