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Theorem reusngf 4633
Description: Restricted existential uniqueness over a singleton. (Contributed by AV, 3-Apr-2023.)
Hypotheses
Ref Expression
rexsngf.1 𝑥𝜓
rexsngf.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
reusngf (𝐴𝑉 → (∃!𝑥 ∈ {𝐴}𝜑𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem reusngf
Dummy variables 𝑤 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfsbc1v 3759 . . 3 𝑥[𝑐 / 𝑥]𝜑
2 nfsbc1v 3759 . . 3 𝑥[𝑤 / 𝑥]𝜑
3 sbceq1a 3750 . . 3 (𝑥 = 𝑤 → (𝜑[𝑤 / 𝑥]𝜑))
4 dfsbcq 3741 . . 3 (𝑤 = 𝑐 → ([𝑤 / 𝑥]𝜑[𝑐 / 𝑥]𝜑))
51, 2, 3, 4reu8nf 3833 . 2 (∃!𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥 ∈ {𝐴} (𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝑥 = 𝑐)))
6 rexsngf.1 . . . . 5 𝑥𝜓
7 nfcv 2907 . . . . . 6 𝑥{𝐴}
8 nfv 1917 . . . . . . 7 𝑥 𝐴 = 𝑐
91, 8nfim 1899 . . . . . 6 𝑥([𝑐 / 𝑥]𝜑𝐴 = 𝑐)
107, 9nfralw 3294 . . . . 5 𝑥𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐)
116, 10nfan 1902 . . . 4 𝑥(𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐))
12 rexsngf.2 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
13 eqeq1 2740 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝑐𝐴 = 𝑐))
1413imbi2d 340 . . . . . 6 (𝑥 = 𝐴 → (([𝑐 / 𝑥]𝜑𝑥 = 𝑐) ↔ ([𝑐 / 𝑥]𝜑𝐴 = 𝑐)))
1514ralbidv 3174 . . . . 5 (𝑥 = 𝐴 → (∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝑥 = 𝑐) ↔ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐)))
1612, 15anbi12d 631 . . . 4 (𝑥 = 𝐴 → ((𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝑥 = 𝑐)) ↔ (𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐))))
1711, 16rexsngf 4631 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴} (𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝑥 = 𝑐)) ↔ (𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐))))
18 nfv 1917 . . . . . 6 𝑐([𝐴 / 𝑥]𝜑𝐴 = 𝐴)
19 dfsbcq 3741 . . . . . . 7 (𝑐 = 𝐴 → ([𝑐 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
20 eqeq2 2748 . . . . . . 7 (𝑐 = 𝐴 → (𝐴 = 𝑐𝐴 = 𝐴))
2119, 20imbi12d 344 . . . . . 6 (𝑐 = 𝐴 → (([𝑐 / 𝑥]𝜑𝐴 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑𝐴 = 𝐴)))
2218, 21ralsngf 4632 . . . . 5 (𝐴𝑉 → (∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑𝐴 = 𝐴)))
2322anbi2d 629 . . . 4 (𝐴𝑉 → ((𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐)) ↔ (𝜓 ∧ ([𝐴 / 𝑥]𝜑𝐴 = 𝐴))))
24 eqidd 2737 . . . . 5 ([𝐴 / 𝑥]𝜑𝐴 = 𝐴)
2524biantru 530 . . . 4 (𝜓 ↔ (𝜓 ∧ ([𝐴 / 𝑥]𝜑𝐴 = 𝐴)))
2623, 25bitr4di 288 . . 3 (𝐴𝑉 → ((𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐)) ↔ 𝜓))
2717, 26bitrd 278 . 2 (𝐴𝑉 → (∃𝑥 ∈ {𝐴} (𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝑥 = 𝑐)) ↔ 𝜓))
285, 27bitrid 282 1 (𝐴𝑉 → (∃!𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wnf 1785  wcel 2106  wral 3064  wrex 3073  ∃!wreu 3351  [wsbc 3739  {csn 4586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-v 3447  df-sbc 3740  df-sn 4587
This theorem is referenced by:  reusng  4636  rmosn  4680
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