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Theorem reuxfr1d 3716
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr1ds 3717. (Contributed by Thierry Arnoux, 7-Apr-2017.)
Hypotheses
Ref Expression
reuxfr1d.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
reuxfr1d.2 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
reuxfr1d.3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
reuxfr1d (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Distinct variable groups:   𝑥,𝑦,𝜑   𝜓,𝑦   𝜒,𝑥   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfr1d
StepHypRef Expression
1 reuxfr1d.2 . . . . . 6 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
2 reurex 3404 . . . . . 6 (∃!𝑦𝐶 𝑥 = 𝐴 → ∃𝑦𝐶 𝑥 = 𝐴)
31, 2syl 17 . . . . 5 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
43biantrurd 536 . . . 4 ((𝜑𝑥𝐵) → (𝜓 ↔ (∃𝑦𝐶 𝑥 = 𝐴𝜓)))
5 r19.41v 3328 . . . . . 6 (∃𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ (∃𝑦𝐶 𝑥 = 𝐴𝜓))
6 reuxfr1d.3 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
76pm5.32da 582 . . . . . . 7 (𝜑 → ((𝑥 = 𝐴𝜓) ↔ (𝑥 = 𝐴𝜒)))
87rexbidv 3283 . . . . . 6 (𝜑 → (∃𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ ∃𝑦𝐶 (𝑥 = 𝐴𝜒)))
95, 8bitr3id 288 . . . . 5 (𝜑 → ((∃𝑦𝐶 𝑥 = 𝐴𝜓) ↔ ∃𝑦𝐶 (𝑥 = 𝐴𝜒)))
109adantr 484 . . . 4 ((𝜑𝑥𝐵) → ((∃𝑦𝐶 𝑥 = 𝐴𝜓) ↔ ∃𝑦𝐶 (𝑥 = 𝐴𝜒)))
114, 10bitrd 282 . . 3 ((𝜑𝑥𝐵) → (𝜓 ↔ ∃𝑦𝐶 (𝑥 = 𝐴𝜒)))
1211reubidva 3369 . 2 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜒)))
13 reuxfr1d.1 . . 3 ((𝜑𝑦𝐶) → 𝐴𝐵)
14 reurmo 3406 . . . 4 (∃!𝑦𝐶 𝑥 = 𝐴 → ∃*𝑦𝐶 𝑥 = 𝐴)
151, 14syl 17 . . 3 ((𝜑𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)
1613, 15reuxfrd 3714 . 2 (𝜑 → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜒) ↔ ∃!𝑦𝐶 𝜒))
1712, 16bitrd 282 1 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wrex 3131  ∃!wreu 3132  ∃*wrmo 3133
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-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-cleq 2815  df-clel 2894  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138
This theorem is referenced by:  reuxfr1ds  3717  rmoxfrd  30262  fcnvgreu  30426  reuf1odnf  43606  reuf1od  43607
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