Theorem 3rspcedvdw 3640

Description: Triple application of rspcedvdw 3625. (Contributed by SN, 20-Aug-2024.)

Hypotheses

Ref	Expression
3rspcedvdw.1	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
3rspcedvdw.2	⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))
3rspcedvdw.3	⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏))
3rspcedvdw.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
3rspcedvdw.b	⊢ (𝜑 → 𝐵 ∈ 𝑌)
3rspcedvdw.c	⊢ (𝜑 → 𝐶 ∈ 𝑍)
3rspcedvdw.4	⊢ (𝜑 → 𝜏)

Assertion

Ref	Expression
3rspcedvdw	⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∃𝑧 ∈ 𝑍 𝜓)

Distinct variable groups:   𝜒,𝑥   𝜃,𝑦   𝜏,𝑧   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝑋   𝑥,𝑌,𝑦   𝑥,𝑍,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑦,𝑧)   𝜃(𝑥,𝑧)   𝜏(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝑋(𝑦,𝑧)   𝑌(𝑧)

Proof of Theorem 3rspcedvdw

Step	Hyp	Ref	Expression
1		3rspcedvdw.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
2		3rspcedvdw.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
3		3rspcedvdw.c	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
4		3rspcedvdw.4	. 2 ⊢ (𝜑 → 𝜏)
5		3rspcedvdw.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
6		3rspcedvdw.2	. . 3 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))
7		3rspcedvdw.3	. . 3 ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏))
8	5, 6, 7	rspc3ev 3639	. 2 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝜏) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∃𝑧 ∈ 𝑍 𝜓)
9	1, 2, 3, 4, 8	syl31anc 1371	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∃𝑧 ∈ 𝑍 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1535   ∈ wcel 2104  ∃wrex 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1087  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067

This theorem is referenced by:  dffltz  42575  nna4b4nsq  42601  grlimgrtri  47821