Theorem 2rspcedvdw

Description: Double application of rspcedvdw . (Contributed by SN, 24-Aug-2024)

	Ref	Expression
Hypotheses	2rspcedvdw.1	⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) )
	2rspcedvdw.2	⊢ ( 𝑦 = 𝐵 → ( 𝜒 ↔ 𝜃 ) )
	2rspcedvdw.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
	2rspcedvdw.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑌 )
	2rspcedvdw.3	⊢ ( 𝜑 → 𝜃 )
Assertion	2rspcedvdw	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑋 ∃ 𝑦 ∈ 𝑌 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		2rspcedvdw.1	⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) )
2		2rspcedvdw.2	⊢ ( 𝑦 = 𝐵 → ( 𝜒 ↔ 𝜃 ) )
3		2rspcedvdw.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
4		2rspcedvdw.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑌 )
5		2rspcedvdw.3	⊢ ( 𝜑 → 𝜃 )
6	1 2	rspc2ev	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝜃 ) → ∃ 𝑥 ∈ 𝑋 ∃ 𝑦 ∈ 𝑌 𝜓 )
7	3 4 5 6	syl3anc	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑋 ∃ 𝑦 ∈ 𝑌 𝜓 )