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Theorem rmoxfrd 29666
Description: Transfer "at most one" restricted quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
rmoxfrd.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
rmoxfrd.2 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
rmoxfrd.3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rmoxfrd (𝜑 → (∃*𝑥𝐵 𝜓 ↔ ∃*𝑦𝐶 𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem rmoxfrd
StepHypRef Expression
1 rmoxfrd.1 . . . . . 6 ((𝜑𝑦𝐶) → 𝐴𝐵)
2 rmoxfrd.2 . . . . . . 7 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
3 reurex 3308 . . . . . . 7 (∃!𝑦𝐶 𝑥 = 𝐴 → ∃𝑦𝐶 𝑥 = 𝐴)
42, 3syl 17 . . . . . 6 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
5 rmoxfrd.3 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
61, 4, 5rexxfrd 5009 . . . . 5 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
7 df-rex 3066 . . . . 5 (∃𝑥𝐵 𝜓 ↔ ∃𝑥(𝑥𝐵𝜓))
8 df-rex 3066 . . . . 5 (∃𝑦𝐶 𝜒 ↔ ∃𝑦(𝑦𝐶𝜒))
96, 7, 83bitr3g 302 . . . 4 (𝜑 → (∃𝑥(𝑥𝐵𝜓) ↔ ∃𝑦(𝑦𝐶𝜒)))
101, 2, 5reuxfr4d 29663 . . . . 5 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
11 df-reu 3067 . . . . 5 (∃!𝑥𝐵 𝜓 ↔ ∃!𝑥(𝑥𝐵𝜓))
12 df-reu 3067 . . . . 5 (∃!𝑦𝐶 𝜒 ↔ ∃!𝑦(𝑦𝐶𝜒))
1310, 11, 123bitr3g 302 . . . 4 (𝜑 → (∃!𝑥(𝑥𝐵𝜓) ↔ ∃!𝑦(𝑦𝐶𝜒)))
149, 13imbi12d 333 . . 3 (𝜑 → ((∃𝑥(𝑥𝐵𝜓) → ∃!𝑥(𝑥𝐵𝜓)) ↔ (∃𝑦(𝑦𝐶𝜒) → ∃!𝑦(𝑦𝐶𝜒))))
15 df-mo 2622 . . 3 (∃*𝑥(𝑥𝐵𝜓) ↔ (∃𝑥(𝑥𝐵𝜓) → ∃!𝑥(𝑥𝐵𝜓)))
16 df-mo 2622 . . 3 (∃*𝑦(𝑦𝐶𝜒) ↔ (∃𝑦(𝑦𝐶𝜒) → ∃!𝑦(𝑦𝐶𝜒)))
1714, 15, 163bitr4g 303 . 2 (𝜑 → (∃*𝑥(𝑥𝐵𝜓) ↔ ∃*𝑦(𝑦𝐶𝜒)))
18 df-rmo 3068 . 2 (∃*𝑥𝐵 𝜓 ↔ ∃*𝑥(𝑥𝐵𝜓))
19 df-rmo 3068 . 2 (∃*𝑦𝐶 𝜒 ↔ ∃*𝑦(𝑦𝐶𝜒))
2017, 18, 193bitr4g 303 1 (𝜑 → (∃*𝑥𝐵 𝜓 ↔ ∃*𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wex 1851  wcel 2144  ∃!weu 2617  ∃*wmo 2618  wrex 3061  ∃!wreu 3062  ∃*wrmo 3063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-v 3351
This theorem is referenced by:  disjrdx  29736
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