Theorem vtocl3ga

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995)

	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		nfcv	⊢ Ⅎ 𝑥 𝐴
6		nfcv	⊢ Ⅎ 𝑦 𝐴
7		nfcv	⊢ Ⅎ 𝑧 𝐴
8		nfcv	⊢ Ⅎ 𝑦 𝐵
9		nfcv	⊢ Ⅎ 𝑧 𝐵
10		nfcv	⊢ Ⅎ 𝑧 𝐶
11		nfv	⊢ Ⅎ 𝑥 𝜓
12		nfv	⊢ Ⅎ 𝑦 𝜒
13		nfv	⊢ Ⅎ 𝑧 𝜃
14	5 6 7 8 9 10 11 12 13 1 2 3 4	vtocl3gaf	⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → 𝜃 )