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Metamath Proof Explorer

Theorem vtocl3ga

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995) Reduce axiom usage. (Revised by GG, 3-Oct-2024) (Proof shortened by Wolf Lammen, 31-May-2025)

		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	3	imbi2d	⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ) → 𝜒 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ) → 𝜃 ) ) )
6	1	imbi2d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑧 ∈ 𝑆 → 𝜑 ) ↔ ( 𝑧 ∈ 𝑆 → 𝜓 ) ) )
7	2	imbi2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝑧 ∈ 𝑆 → 𝜓 ) ↔ ( 𝑧 ∈ 𝑆 → 𝜒 ) ) )
8	4	3expia	⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ) → ( 𝑧 ∈ 𝑆 → 𝜑 ) )
9	6 7 8	vtocl2ga	⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ) → ( 𝑧 ∈ 𝑆 → 𝜒 ) )
10	9	com12	⊢ ( 𝑧 ∈ 𝑆 → ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ) → 𝜒 ) )
11	5 10	vtoclga	⊢ ( 𝐶 ∈ 𝑆 → ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ) → 𝜃 ) )
12	11	impcom	⊢ ( ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ) ∧ 𝐶 ∈ 𝑆 ) → 𝜃 )
13	12	3impa	⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → 𝜃 )