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Theorem reuxfrd 5042
 Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 5044 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.)
Hypotheses
Ref Expression
reuxfrd.1 ((𝜑𝑦𝐵) → 𝐴𝐵)
reuxfrd.2 ((𝜑𝑥𝐵) → ∃!𝑦𝐵 𝑥 = 𝐴)
reuxfrd.3 (𝑥 = 𝐴 → (𝜓𝜒))
Assertion
Ref Expression
reuxfrd (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐵 𝜒))
Distinct variable groups:   𝑥,𝑦,𝜑   𝜓,𝑦   𝜒,𝑥   𝑥,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfrd
StepHypRef Expression
1 reuxfrd.2 . . . . . 6 ((𝜑𝑥𝐵) → ∃!𝑦𝐵 𝑥 = 𝐴)
2 reurex 3299 . . . . . 6 (∃!𝑦𝐵 𝑥 = 𝐴 → ∃𝑦𝐵 𝑥 = 𝐴)
31, 2syl 17 . . . . 5 ((𝜑𝑥𝐵) → ∃𝑦𝐵 𝑥 = 𝐴)
43biantrurd 530 . . . 4 ((𝜑𝑥𝐵) → (𝜓 ↔ (∃𝑦𝐵 𝑥 = 𝐴𝜓)))
5 r19.41v 3227 . . . . 5 (∃𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ (∃𝑦𝐵 𝑥 = 𝐴𝜓))
6 reuxfrd.3 . . . . . . 7 (𝑥 = 𝐴 → (𝜓𝜒))
76pm5.32i 672 . . . . . 6 ((𝑥 = 𝐴𝜓) ↔ (𝑥 = 𝐴𝜒))
87rexbii 3179 . . . . 5 (∃𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ ∃𝑦𝐵 (𝑥 = 𝐴𝜒))
95, 8bitr3i 266 . . . 4 ((∃𝑦𝐵 𝑥 = 𝐴𝜓) ↔ ∃𝑦𝐵 (𝑥 = 𝐴𝜒))
104, 9syl6bb 276 . . 3 ((𝜑𝑥𝐵) → (𝜓 ↔ ∃𝑦𝐵 (𝑥 = 𝐴𝜒)))
1110reubidva 3264 . 2 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜒)))
12 reuxfrd.1 . . 3 ((𝜑𝑦𝐵) → 𝐴𝐵)
13 reurmo 3300 . . . 4 (∃!𝑦𝐵 𝑥 = 𝐴 → ∃*𝑦𝐵 𝑥 = 𝐴)
141, 13syl 17 . . 3 ((𝜑𝑥𝐵) → ∃*𝑦𝐵 𝑥 = 𝐴)
1512, 14reuxfr2d 5040 . 2 (𝜑 → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜒) ↔ ∃!𝑦𝐵 𝜒))
1611, 15bitrd 268 1 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐵 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:  reuxfr  5043  riotaxfrd  6805
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