Theorem vtocl3ga

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995) Reduce axiom usage. (Revised by Gino Giotto, 3-Oct-2024)

	Ref	Expression
Hypotheses	vtocl3ga.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	vtocl3ga.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
	vtocl3ga.3	⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
	vtocl3ga.4	⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜑 )
Assertion	vtocl3ga	⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		vtocl3ga.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		vtocl3ga.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
3		vtocl3ga.3	⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
4		vtocl3ga.4	⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜑 )
5		eleq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝐷 ↔ 𝐴 ∈ 𝐷 ) )
6	5	3anbi1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) ) )
7	6 1	imbi12d	⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜑 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜓 ) ) )
8		eleq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∈ 𝑅 ↔ 𝐵 ∈ 𝑅 ) )
9	8	3anbi2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) ) )
10	9 2	imbi12d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜓 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜒 ) ) )
11		eleq1	⊢ ( 𝑧 = 𝐶 → ( 𝑧 ∈ 𝑆 ↔ 𝐶 ∈ 𝑆 ) )
12	11	3anbi3d	⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) ) )
13	12 3	imbi12d	⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜒 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → 𝜃 ) ) )
14	7 10 13 4	vtocl3g	⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → 𝜃 ) )
15	14	pm2.43i	⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → 𝜃 )