Theorem 2rspcedvdw 3636

Description: Double application of rspcedvdw 3625. (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 3635	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝜃) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓)
7	1, 2, 3, 6	syl3anc 1373	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ∃wrex 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071

This theorem is referenced by:  gsumwun  33068  elrgspnlem2  33247  elrspunsn  33457  posbezout  42101  flt4lem7  42669  nna4b4nsq  42670  usgrgrtrirex  47917  gpg3kgrtriex  48045