Theorem 2rspcedvdw 3593

Description: Double application of rspcedvdw 3582. (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 3592	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝜃) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓)
7	1, 2, 3, 6	syl3anc 1398	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∃wrex 3088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089

This theorem is used by:  rspc3ev  3596  2sqnn0  27670  z12addscl  28738  z12shalf  28741  z12zsodd  28743  elq2  33267  gsumwun  33501  elrgspnlem2  33668  elrspunsn  33842  posbezout  42951  flt4lem7  43490  nna4b4nsq  43491  nprmmul2  48413  usgrgrtrirex  48851  gpg3kgrtriex  48990  grlimedgnedg  49032