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Theorem reuxfr3d 29637
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr2d 5040. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
reuxfr3d.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
reuxfr3d.2 ((𝜑𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)
Assertion
Ref Expression
reuxfr3d (𝜑 → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐶 𝜓))
Distinct variable groups:   𝑥,𝑦,𝜑   𝜓,𝑥   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦
Allowed substitution hints:   𝜓(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfr3d
StepHypRef Expression
1 reuxfr3d.2 . . . . . . 7 ((𝜑𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)
2 rmoan 3547 . . . . . . 7 (∃*𝑦𝐶 𝑥 = 𝐴 → ∃*𝑦𝐶 (𝜓𝑥 = 𝐴))
31, 2syl 17 . . . . . 6 ((𝜑𝑥𝐵) → ∃*𝑦𝐶 (𝜓𝑥 = 𝐴))
4 ancom 465 . . . . . . 7 ((𝜓𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝜓))
54rmobii 3272 . . . . . 6 (∃*𝑦𝐶 (𝜓𝑥 = 𝐴) ↔ ∃*𝑦𝐶 (𝑥 = 𝐴𝜓))
63, 5sylib 208 . . . . 5 ((𝜑𝑥𝐵) → ∃*𝑦𝐶 (𝑥 = 𝐴𝜓))
76ralrimiva 3104 . . . 4 (𝜑 → ∀𝑥𝐵 ∃*𝑦𝐶 (𝑥 = 𝐴𝜓))
8 2reuswap 3551 . . . 4 (∀𝑥𝐵 ∃*𝑦𝐶 (𝑥 = 𝐴𝜓) → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓) → ∃!𝑦𝐶𝑥𝐵 (𝑥 = 𝐴𝜓)))
97, 8syl 17 . . 3 (𝜑 → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓) → ∃!𝑦𝐶𝑥𝐵 (𝑥 = 𝐴𝜓)))
10 2reuswap2 29636 . . . 4 (∀𝑦𝐶 ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)) → (∃!𝑦𝐶𝑥𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓)))
11 moeq 3523 . . . . . . 7 ∃*𝑥 𝑥 = 𝐴
1211moani 2663 . . . . . 6 ∃*𝑥((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴)
13 ancom 465 . . . . . . . 8 (((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ (𝑥𝐵𝜓)))
14 an12 873 . . . . . . . 8 ((𝑥 = 𝐴 ∧ (𝑥𝐵𝜓)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1513, 14bitri 264 . . . . . . 7 (((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1615mobii 2630 . . . . . 6 (∃*𝑥((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1712, 16mpbi 220 . . . . 5 ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓))
1817a1i 11 . . . 4 (𝑦𝐶 → ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1910, 18mprg 3064 . . 3 (∃!𝑦𝐶𝑥𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓))
209, 19impbid1 215 . 2 (𝜑 → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐶𝑥𝐵 (𝑥 = 𝐴𝜓)))
21 reuxfr3d.1 . . . 4 ((𝜑𝑦𝐶) → 𝐴𝐵)
22 biidd 252 . . . . 5 (𝑥 = 𝐴 → (𝜓𝜓))
2322ceqsrexv 3475 . . . 4 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ 𝜓))
2421, 23syl 17 . . 3 ((𝜑𝑦𝐶) → (∃𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ 𝜓))
2524reubidva 3264 . 2 (𝜑 → (∃!𝑦𝐶𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐶 𝜓))
2620, 25bitrd 268 1 (𝜑 → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐶 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  ∃*wmo 2608  wral 3050  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:  reuxfr4d  29638
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