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Theorem rmoxfrdOLD 28534
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.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
rmoxfrd.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
rmoxfrd.2 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
rmoxfrd.3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rmoxfrdOLD (𝜑 → (∃*𝑥(𝑥𝐵𝜓) ↔ ∃*𝑦(𝑦𝐶𝜒)))
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 rmoxfrdOLD
StepHypRef Expression
1 rmoxfrd.1 . . . . 5 ((𝜑𝑦𝐶) → 𝐴𝐵)
2 rmoxfrd.2 . . . . . 6 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
3 reurex 3041 . . . . . 6 (∃!𝑦𝐶 𝑥 = 𝐴 → ∃𝑦𝐶 𝑥 = 𝐴)
42, 3syl 17 . . . . 5 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
5 rmoxfrd.3 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
61, 4, 5rexxfrd 4706 . . . 4 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
7 df-rex 2806 . . . 4 (∃𝑥𝐵 𝜓 ↔ ∃𝑥(𝑥𝐵𝜓))
8 df-rex 2806 . . . 4 (∃𝑦𝐶 𝜒 ↔ ∃𝑦(𝑦𝐶𝜒))
96, 7, 83bitr3g 300 . . 3 (𝜑 → (∃𝑥(𝑥𝐵𝜓) ↔ ∃𝑦(𝑦𝐶𝜒)))
101, 2, 5reuxfr4d 28532 . . . 4 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
11 df-reu 2807 . . . 4 (∃!𝑥𝐵 𝜓 ↔ ∃!𝑥(𝑥𝐵𝜓))
12 df-reu 2807 . . . 4 (∃!𝑦𝐶 𝜒 ↔ ∃!𝑦(𝑦𝐶𝜒))
1310, 11, 123bitr3g 300 . . 3 (𝜑 → (∃!𝑥(𝑥𝐵𝜓) ↔ ∃!𝑦(𝑦𝐶𝜒)))
149, 13imbi12d 332 . 2 (𝜑 → ((∃𝑥(𝑥𝐵𝜓) → ∃!𝑥(𝑥𝐵𝜓)) ↔ (∃𝑦(𝑦𝐶𝜒) → ∃!𝑦(𝑦𝐶𝜒))))
15 df-mo 2367 . 2 (∃*𝑥(𝑥𝐵𝜓) ↔ (∃𝑥(𝑥𝐵𝜓) → ∃!𝑥(𝑥𝐵𝜓)))
16 df-mo 2367 . 2 (∃*𝑦(𝑦𝐶𝜒) ↔ (∃𝑦(𝑦𝐶𝜒) → ∃!𝑦(𝑦𝐶𝜒)))
1714, 15, 163bitr4g 301 1 (𝜑 → (∃*𝑥(𝑥𝐵𝜓) ↔ ∃*𝑦(𝑦𝐶𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1938  ∃!weu 2362  ∃*wmo 2363  wrex 2801  ∃!wreu 2802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-v 3079
This theorem is referenced by: (None)
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