Theorem rexlimdvvva

Description: Inference from Theorem 19.23 of Margaris p. 90, for three restricted quantifiers. (Contributed by AV, 23-Aug-2025)

	Ref	Expression
Hypothesis	rexlimdvvva.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝜓 → 𝜒 ) )
Assertion	rexlimdvvva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rexlimdvvva.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝜓 → 𝜒 ) )
2		df-3an	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐶 ) )
3	1	ex	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → ( 𝜓 → 𝜒 ) ) )
4	2 3	biimtrrid	⊢ ( 𝜑 → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐶 ) → ( 𝜓 → 𝜒 ) ) )
5	4	expdimp	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑧 ∈ 𝐶 → ( 𝜓 → 𝜒 ) ) )
6	5	rexlimdv	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( ∃ 𝑧 ∈ 𝐶 𝜓 → 𝜒 ) )
7	6	rexlimdvva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝜓 → 𝜒 ) )