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Theorem reusv1 5015
 Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
Assertion
Ref Expression
reusv1 (∃𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem reusv1
StepHypRef Expression
1 nfra1 3079 . . . 4 𝑦𝑦𝐵 (𝜑𝑥 = 𝐶)
21nfmo 2624 . . 3 𝑦∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)
3 rsp 3067 . . . . . 6 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → (𝑦𝐵 → (𝜑𝑥 = 𝐶)))
43com3l 89 . . . . 5 (𝑦𝐵 → (𝜑 → (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶)))
54alrimdv 2006 . . . 4 (𝑦𝐵 → (𝜑 → ∀𝑥(∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶)))
6 mo2icl 3526 . . . 4 (∀𝑥(∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶) → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶))
75, 6syl6 35 . . 3 (𝑦𝐵 → (𝜑 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)))
82, 7rexlimi 3162 . 2 (∃𝑦𝐵 𝜑 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶))
9 mormo 3297 . 2 (∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶) → ∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
10 reu5 3298 . . 3 (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
1110rbaib 985 . 2 (∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
128, 9, 113syl 18 1 (∃𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1630   = 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:  cdleme25c  36145  cdleme29c  36166  cdlemefrs29cpre1  36188  cdlemk29-3  36701  cdlemkid5  36725  dihlsscpre  37025  mapdh9a  37581  mapdh9aOLDN  37582
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