Theorem fucofulem1 48957

Description: Lemma for proving functor theorems. (Contributed by Zhi Wang, 25-Sep-2025.)

Hypotheses

Ref	Expression
fucofulem1.1	⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))
fucofulem1.2	⊢ ((𝜑 ∧ (𝜃 ∧ 𝜏)) → 𝜂)
fucofulem1.3	⊢ 𝜒
fucofulem1.4	⊢ ((𝜑 ∧ 𝜂) → 𝜃)
fucofulem1.5	⊢ ((𝜑 ∧ 𝜂) → 𝜏)

Assertion

Ref	Expression
fucofulem1	⊢ (𝜑 → (𝜓 ↔ 𝜂))

Proof of Theorem fucofulem1

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2		fucofulem1.1	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))
3	2	biimpa 476	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃 ∧ 𝜏))
4	3	simp2d 1143	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
5	3	simp3d 1144	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
6		fucofulem1.2	. . 3 ⊢ ((𝜑 ∧ (𝜃 ∧ 𝜏)) → 𝜂)
7	1, 4, 5, 6	syl12anc 836	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜂)
8		simpl 482	. . 3 ⊢ ((𝜑 ∧ 𝜂) → 𝜑)
9		fucofulem1.3	. . . 4 ⊢ 𝜒
10	9	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝜂) → 𝜒)
11		fucofulem1.4	. . 3 ⊢ ((𝜑 ∧ 𝜂) → 𝜃)
12		fucofulem1.5	. . 3 ⊢ ((𝜑 ∧ 𝜂) → 𝜏)
13	2	biimpar 477	. . 3 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃 ∧ 𝜏)) → 𝜓)
14	8, 10, 11, 12, 13	syl13anc 1373	. 2 ⊢ ((𝜑 ∧ 𝜂) → 𝜓)
15	7, 14	impbida 800	1 ⊢ (𝜑 → (𝜓 ↔ 𝜂))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088

This theorem is referenced by: (None)