Theorem 2rspcedvdw 40177

Description: Double application of rspcedvdw 40176. (Contributed by SN, 24-Aug-2024.)

Hypotheses

Ref	Expression
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

Ref	Expression
2rspcedvdw	⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝑋   𝑥,𝑌,𝑦   𝜒,𝑥   𝜃,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)   𝜃(𝑥)   𝐵(𝑥)   𝑋(𝑦)

Proof of Theorem 2rspcedvdw

Step	Hyp	Ref	Expression
1		2rspcedvdw.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
2		2rspcedvdw.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
3		2rspcedvdw.3	. 2 ⊢ (𝜑 → 𝜃)
4		2rspcedvdw.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
5		2rspcedvdw.2	. . 3 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))
6	4, 5	rspc2ev 3573	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝜃) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓)
7	1, 2, 3, 6	syl3anc 1370	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2110  ∃wrex 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072

This theorem is referenced by:  flt4lem7  40493  nna4b4nsq  40494