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Theorem reuxfrd 5040
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 5042 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 3297 . . . . . 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 3225 . . . . 5 (∃𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ (∃𝑦𝐵 𝑥 = 𝐴𝜓))
6 reuxfrd.3 . . . . . . 7 (𝑥 = 𝐴 → (𝜓𝜒))
76pm5.32i 672 . . . . . 6 ((𝑥 = 𝐴𝜓) ↔ (𝑥 = 𝐴𝜒))
87rexbii 3177 . . . . 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 3262 . 2 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜒)))
12 reuxfrd.1 . . 3 ((𝜑𝑦𝐵) → 𝐴𝐵)
13 reurmo 3298 . . . 4 (∃!𝑦𝐵 𝑥 = 𝐴 → ∃*𝑦𝐵 𝑥 = 𝐴)
141, 13syl 17 . . 3 ((𝜑𝑥𝐵) → ∃*𝑦𝐵 𝑥 = 𝐴)
1512, 14reuxfr2d 5038 . 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 1630  wcel 2137  wrex 3049  ∃!wreu 3050  ∃*wrmo 3051
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 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-v 3340
This theorem is referenced by:  reuxfr  5041  riotaxfrd  6803
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