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Theorem catprslem 45819
Description: Lemma for catprs 45820. (Contributed by Zhi Wang, 18-Sep-2024.)
Hypotheses
Ref Expression
catprs.1 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
catprslem.x (𝜑𝑋𝐵)
catprslem.y (𝜑𝑌𝐵)
Assertion
Ref Expression
catprslem (𝜑 → (𝑋 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
Distinct variable groups:   𝑥, ,𝑦   𝑥,𝐵,𝑦   𝑥,𝐻,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem catprslem
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catprs.1 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
2 breq1 5043 . . . . 5 (𝑥 = 𝑧 → (𝑥 𝑦𝑧 𝑦))
3 oveq1 7189 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
43neeq1d 2994 . . . . 5 (𝑥 = 𝑧 → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑦) ≠ ∅))
52, 4bibi12d 349 . . . 4 (𝑥 = 𝑧 → ((𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅) ↔ (𝑧 𝑦 ↔ (𝑧𝐻𝑦) ≠ ∅)))
6 breq2 5044 . . . . 5 (𝑦 = 𝑤 → (𝑧 𝑦𝑧 𝑤))
7 oveq2 7190 . . . . . 6 (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
87neeq1d 2994 . . . . 5 (𝑦 = 𝑤 → ((𝑧𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
96, 8bibi12d 349 . . . 4 (𝑦 = 𝑤 → ((𝑧 𝑦 ↔ (𝑧𝐻𝑦) ≠ ∅) ↔ (𝑧 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)))
105, 9cbvral2vw 3363 . . 3 (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅) ↔ ∀𝑧𝐵𝑤𝐵 (𝑧 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
111, 10sylib 221 . 2 (𝜑 → ∀𝑧𝐵𝑤𝐵 (𝑧 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
12 catprslem.x . . 3 (𝜑𝑋𝐵)
13 catprslem.y . . 3 (𝜑𝑌𝐵)
14 breq12 5045 . . . . 5 ((𝑧 = 𝑋𝑤 = 𝑌) → (𝑧 𝑤𝑋 𝑌))
15 oveq12 7191 . . . . . 6 ((𝑧 = 𝑋𝑤 = 𝑌) → (𝑧𝐻𝑤) = (𝑋𝐻𝑌))
1615neeq1d 2994 . . . . 5 ((𝑧 = 𝑋𝑤 = 𝑌) → ((𝑧𝐻𝑤) ≠ ∅ ↔ (𝑋𝐻𝑌) ≠ ∅))
1714, 16bibi12d 349 . . . 4 ((𝑧 = 𝑋𝑤 = 𝑌) → ((𝑧 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅) ↔ (𝑋 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)))
1817rspc2gv 3538 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑧𝐵𝑤𝐵 (𝑧 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅) → (𝑋 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)))
1912, 13, 18syl2anc 587 . 2 (𝜑 → (∀𝑧𝐵𝑤𝐵 (𝑧 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅) → (𝑋 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)))
2011, 19mpd 15 1 (𝜑 → (𝑋 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wne 2935  wral 3054  c0 4221   class class class wbr 5040  (class class class)co 7182
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 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ne 2936  df-ral 3059  df-v 3402  df-un 3858  df-in 3860  df-ss 3870  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6307  df-fv 6357  df-ov 7185
This theorem is referenced by:  catprs  45820
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