Theorem 2rspcedvdw 3586

Description: Double application of rspcedvdw 3575. (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 3585	. 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 1541   ∈ wcel 2111  ∃wrex 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057

This theorem is referenced by:  zs12addscl  28387  zs12half  28390  zs12zodd  28392  elq2  32794  gsumwun  33045  elrgspnlem2  33210  elrspunsn  33394  posbezout  42192  flt4lem7  42751  nna4b4nsq  42752  usgrgrtrirex  48049  gpg3kgrtriex  48188  grlimedgnedg  48230