Theorem vtocl4g

Description: Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019)

	Ref	Expression
Hypotheses	vtocl4ga.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	vtocl4ga.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
	vtocl4ga.3	⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜌 ) )
	vtocl4ga.4	⊢ ( 𝑤 = 𝐷 → ( 𝜌 ↔ 𝜃 ) )
	vtocl4g.5	⊢ 𝜑
Assertion	vtocl4g	⊢ ( ( ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅 ) ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇 ) ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		vtocl4ga.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		vtocl4ga.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
3		vtocl4ga.3	⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜌 ) )
4		vtocl4ga.4	⊢ ( 𝑤 = 𝐷 → ( 𝜌 ↔ 𝜃 ) )
5		vtocl4g.5	⊢ 𝜑
6	3	imbi2d	⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅 ) → 𝜒 ) ↔ ( ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅 ) → 𝜌 ) ) )
7	4	imbi2d	⊢ ( 𝑤 = 𝐷 → ( ( ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅 ) → 𝜌 ) ↔ ( ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅 ) → 𝜃 ) ) )
8	1 2 5	vtocl2g	⊢ ( ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅 ) → 𝜒 )
9	6 7 8	vtocl2g	⊢ ( ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇 ) → ( ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅 ) → 𝜃 ) )
10	9	impcom	⊢ ( ( ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅 ) ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇 ) ) → 𝜃 )