Theorem rexlimdv3d

Description: An extended version of rexlimdvv to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024)

	Ref	Expression
Hypothesis	rexlimdv3d.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → ( 𝜓 → 𝜒 ) ) )
Assertion	rexlimdv3d	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rexlimdv3d.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → ( 𝜓 → 𝜒 ) ) )
2	1	3expd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → ( 𝑧 ∈ 𝐶 → ( 𝜓 → 𝜒 ) ) ) ) )
3	2	imp4d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝜓 → 𝜒 ) ) )
4	3	expdimp	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → ( 𝜓 → 𝜒 ) ) )
5	4	rexlimdvv	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝜓 → 𝜒 ) )
6	5	rexlimdva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝜓 → 𝜒 ) )