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Theorem cbveud 35069
 Description: Deduction used to change bound variables in an existential uniqueness quantifier, using implicit substitution. (Contributed by ML, 27-Mar-2021.)
Hypotheses
Ref Expression
cbveud.1 𝑥𝜑
cbveud.2 𝑦𝜑
cbveud.3 (𝜑 → Ⅎ𝑦𝜓)
cbveud.4 (𝜑 → Ⅎ𝑥𝜒)
cbveud.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbveud (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem cbveud
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbveud.1 . . . 4 𝑥𝜑
2 cbveud.2 . . . 4 𝑦𝜑
3 cbveud.3 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
4 nfvd 1916 . . . . 5 (𝜑 → Ⅎ𝑦 𝑥 = 𝑧)
53, 4nfbid 1903 . . . 4 (𝜑 → Ⅎ𝑦(𝜓𝑥 = 𝑧))
6 cbveud.4 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
7 nfvd 1916 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
86, 7nfbid 1903 . . . 4 (𝜑 → Ⅎ𝑥(𝜒𝑦 = 𝑧))
9 cbveud.5 . . . . 5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
10 simpr 488 . . . . . . 7 ((𝑥 = 𝑦 ∧ (𝜓𝜒)) → (𝜓𝜒))
11 equequ1 2032 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
1211adantr 484 . . . . . . 7 ((𝑥 = 𝑦 ∧ (𝜓𝜒)) → (𝑥 = 𝑧𝑦 = 𝑧))
1310, 12bibi12d 349 . . . . . 6 ((𝑥 = 𝑦 ∧ (𝜓𝜒)) → ((𝜓𝑥 = 𝑧) ↔ (𝜒𝑦 = 𝑧)))
1413ex 416 . . . . 5 (𝑥 = 𝑦 → ((𝜓𝜒) → ((𝜓𝑥 = 𝑧) ↔ (𝜒𝑦 = 𝑧))))
159, 14sylcom 30 . . . 4 (𝜑 → (𝑥 = 𝑦 → ((𝜓𝑥 = 𝑧) ↔ (𝜒𝑦 = 𝑧))))
161, 2, 5, 8, 15cbv2w 2346 . . 3 (𝜑 → (∀𝑥(𝜓𝑥 = 𝑧) ↔ ∀𝑦(𝜒𝑦 = 𝑧)))
1716exbidv 1922 . 2 (𝜑 → (∃𝑧𝑥(𝜓𝑥 = 𝑧) ↔ ∃𝑧𝑦(𝜒𝑦 = 𝑧)))
18 eu6 2593 . 2 (∃!𝑥𝜓 ↔ ∃𝑧𝑥(𝜓𝑥 = 𝑧))
19 eu6 2593 . 2 (∃!𝑦𝜒 ↔ ∃𝑧𝑦(𝜒𝑦 = 𝑧))
2017, 18, 193bitr4g 317 1 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785  ∃!weu 2587 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-mo 2557  df-eu 2588 This theorem is referenced by:  cbvreud  35070
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