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Theorem reu8 3742
Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
Hypothesis
Ref Expression
rmo4.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
reu8 (∃!𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem reu8
StepHypRef Expression
1 rmo4.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
21cbvreuvw 3402 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
3 reu6 3735 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑦 = 𝑥))
4 dfbi2 474 . . . . 5 ((𝜓𝑦 = 𝑥) ↔ ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)))
54ralbii 3091 . . . 4 (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ ∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)))
6 r19.26 3109 . . . . 5 (∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓)))
7 ancom 460 . . . . . 6 ((𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)) ↔ (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ∧ 𝜑))
8 equcom 2015 . . . . . . . . . 10 (𝑥 = 𝑦𝑦 = 𝑥)
98imbi2i 336 . . . . . . . . 9 ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑦 = 𝑥))
109ralbii 3091 . . . . . . . 8 (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑦 = 𝑥))
1110a1i 11 . . . . . . 7 (𝑥𝐴 → (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑦 = 𝑥)))
12 biimt 360 . . . . . . . 8 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
13 df-ral 3060 . . . . . . . . 9 (∀𝑦𝐴 (𝑦 = 𝑥𝜓) ↔ ∀𝑦(𝑦𝐴 → (𝑦 = 𝑥𝜓)))
14 bi2.04 387 . . . . . . . . . 10 ((𝑦𝐴 → (𝑦 = 𝑥𝜓)) ↔ (𝑦 = 𝑥 → (𝑦𝐴𝜓)))
1514albii 1816 . . . . . . . . 9 (∀𝑦(𝑦𝐴 → (𝑦 = 𝑥𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦𝐴𝜓)))
16 eleq1w 2822 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1716, 1imbi12d 344 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
1817bicomd 223 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑦𝐴𝜓) ↔ (𝑥𝐴𝜑)))
1918equcoms 2017 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑦𝐴𝜓) ↔ (𝑥𝐴𝜑)))
2019equsalvw 2001 . . . . . . . . 9 (∀𝑦(𝑦 = 𝑥 → (𝑦𝐴𝜓)) ↔ (𝑥𝐴𝜑))
2113, 15, 203bitrri 298 . . . . . . . 8 ((𝑥𝐴𝜑) ↔ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))
2212, 21bitrdi 287 . . . . . . 7 (𝑥𝐴 → (𝜑 ↔ ∀𝑦𝐴 (𝑦 = 𝑥𝜓)))
2311, 22anbi12d 632 . . . . . 6 (𝑥𝐴 → ((∀𝑦𝐴 (𝜓𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))))
247, 23bitrid 283 . . . . 5 (𝑥𝐴 → ((𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))))
256, 24bitr4id 290 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)) ↔ (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))))
265, 25bitrid 283 . . 3 (𝑥𝐴 → (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))))
2726rexbiia 3090 . 2 (∃𝑥𝐴𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
282, 3, 273bitri 297 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wcel 2106  wral 3059  wrex 3068  ∃!wreu 3376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-10 2139  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clel 2814  df-ral 3060  df-rex 3069  df-reu 3379
This theorem is referenced by:  reu8nf  3886  updjud  9972  reusq0  15498  reumodprminv  16838  grpinveu  19005  addsq2reu  27499  2sqreulem1  27505  2sqreunnlem1  27508  grpoideu  30538  grpoinveu  30548  cvmlift3lem2  35305  euoreqb  47059  2reu8i  47063  2reuimp0  47064  paireqne  47436  itsclquadeu  48627
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