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Theorem reuxfr4d 29638
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfrd 5042. (Contributed by Thierry Arnoux, 7-Apr-2017.)
Hypotheses
Ref Expression
reuxfr4d.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
reuxfr4d.2 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
reuxfr4d.3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
reuxfr4d (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 reuxfr4d
StepHypRef Expression
1 reuxfr4d.2 . . . . . 6 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
2 reurex 3299 . . . . . 6 (∃!𝑦𝐶 𝑥 = 𝐴 → ∃𝑦𝐶 𝑥 = 𝐴)
31, 2syl 17 . . . . 5 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
43biantrurd 530 . . . 4 ((𝜑𝑥𝐵) → (𝜓 ↔ (∃𝑦𝐶 𝑥 = 𝐴𝜓)))
5 r19.41v 3227 . . . . . 6 (∃𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ (∃𝑦𝐶 𝑥 = 𝐴𝜓))
6 reuxfr4d.3 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
76pm5.32da 676 . . . . . . 7 (𝜑 → ((𝑥 = 𝐴𝜓) ↔ (𝑥 = 𝐴𝜒)))
87rexbidv 3190 . . . . . 6 (𝜑 → (∃𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ ∃𝑦𝐶 (𝑥 = 𝐴𝜒)))
95, 8syl5bbr 274 . . . . 5 (𝜑 → ((∃𝑦𝐶 𝑥 = 𝐴𝜓) ↔ ∃𝑦𝐶 (𝑥 = 𝐴𝜒)))
109adantr 472 . . . 4 ((𝜑𝑥𝐵) → ((∃𝑦𝐶 𝑥 = 𝐴𝜓) ↔ ∃𝑦𝐶 (𝑥 = 𝐴𝜒)))
114, 10bitrd 268 . . 3 ((𝜑𝑥𝐵) → (𝜓 ↔ ∃𝑦𝐶 (𝑥 = 𝐴𝜒)))
1211reubidva 3264 . 2 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜒)))
13 reuxfr4d.1 . . 3 ((𝜑𝑦𝐶) → 𝐴𝐵)
14 reurmo 3300 . . . 4 (∃!𝑦𝐶 𝑥 = 𝐴 → ∃*𝑦𝐶 𝑥 = 𝐴)
151, 14syl 17 . . 3 ((𝜑𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)
1613, 15reuxfr3d 29637 . 2 (𝜑 → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜒) ↔ ∃!𝑦𝐶 𝜒))
1712, 16bitrd 268 1 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  ∃!wreu 3052  ∃*wrmo 3053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-v 3342
This theorem is referenced by:  rmoxfrdOLD  29640  rmoxfrd  29641  fcnvgreu  29781
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