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Theorem reusv3 5042
Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 5034 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
Hypotheses
Ref Expression
reusv3.1 (𝑦 = 𝑧 → (𝜑𝜓))
reusv3.2 (𝑦 = 𝑧𝐶 = 𝐷)
Assertion
Ref Expression
reusv3 (∃𝑦𝐵 (𝜑𝐶𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐶,𝑧   𝑥,𝐷,𝑦   𝜑,𝑥,𝑧   𝜓,𝑥,𝑦   𝑥,𝐴,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑧)   𝐶(𝑦)   𝐷(𝑧)

Proof of Theorem reusv3
StepHypRef Expression
1 reusv3.1 . . . . 5 (𝑦 = 𝑧 → (𝜑𝜓))
2 reusv3.2 . . . . . 6 (𝑦 = 𝑧𝐶 = 𝐷)
32eleq1d 2829 . . . . 5 (𝑦 = 𝑧 → (𝐶𝐴𝐷𝐴))
41, 3anbi12d 624 . . . 4 (𝑦 = 𝑧 → ((𝜑𝐶𝐴) ↔ (𝜓𝐷𝐴)))
54cbvrexv 3320 . . 3 (∃𝑦𝐵 (𝜑𝐶𝐴) ↔ ∃𝑧𝐵 (𝜓𝐷𝐴))
6 nfra2 3093 . . . . 5 𝑧𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)
7 nfv 2009 . . . . 5 𝑧𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)
86, 7nfim 1995 . . . 4 𝑧(∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
9 risset 3209 . . . . . 6 (𝐷𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐷)
10 ralcom 3245 . . . . . . . . . . . . . 14 (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧𝐵𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
11 impexp 441 . . . . . . . . . . . . . . . . . 18 (((𝜑𝜓) → 𝐶 = 𝐷) ↔ (𝜑 → (𝜓𝐶 = 𝐷)))
12 bi2.04 377 . . . . . . . . . . . . . . . . . 18 ((𝜑 → (𝜓𝐶 = 𝐷)) ↔ (𝜓 → (𝜑𝐶 = 𝐷)))
1311, 12bitri 266 . . . . . . . . . . . . . . . . 17 (((𝜑𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → (𝜑𝐶 = 𝐷)))
1413ralbii 3127 . . . . . . . . . . . . . . . 16 (∀𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑦𝐵 (𝜓 → (𝜑𝐶 = 𝐷)))
15 r19.21v 3107 . . . . . . . . . . . . . . . 16 (∀𝑦𝐵 (𝜓 → (𝜑𝐶 = 𝐷)) ↔ (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
1614, 15bitri 266 . . . . . . . . . . . . . . 15 (∀𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
1716ralbii 3127 . . . . . . . . . . . . . 14 (∀𝑧𝐵𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧𝐵 (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
1810, 17bitri 266 . . . . . . . . . . . . 13 (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧𝐵 (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
19 rsp 3076 . . . . . . . . . . . . 13 (∀𝑧𝐵 (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)) → (𝑧𝐵 → (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))))
2018, 19sylbi 208 . . . . . . . . . . . 12 (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → (𝑧𝐵 → (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))))
2120com3l 89 . . . . . . . . . . 11 (𝑧𝐵 → (𝜓 → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))))
2221imp31 408 . . . . . . . . . 10 (((𝑧𝐵𝜓) ∧ ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)) → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))
23 eqeq1 2769 . . . . . . . . . . . . 13 (𝑥 = 𝐷 → (𝑥 = 𝐶𝐷 = 𝐶))
24 eqcom 2772 . . . . . . . . . . . . 13 (𝐷 = 𝐶𝐶 = 𝐷)
2523, 24syl6bb 278 . . . . . . . . . . . 12 (𝑥 = 𝐷 → (𝑥 = 𝐶𝐶 = 𝐷))
2625imbi2d 331 . . . . . . . . . . 11 (𝑥 = 𝐷 → ((𝜑𝑥 = 𝐶) ↔ (𝜑𝐶 = 𝐷)))
2726ralbidv 3133 . . . . . . . . . 10 (𝑥 = 𝐷 → (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
2822, 27syl5ibrcom 238 . . . . . . . . 9 (((𝑧𝐵𝜓) ∧ ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)) → (𝑥 = 𝐷 → ∀𝑦𝐵 (𝜑𝑥 = 𝐶)))
2928reximdv 3162 . . . . . . . 8 (((𝑧𝐵𝜓) ∧ ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)) → (∃𝑥𝐴 𝑥 = 𝐷 → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
3029ex 401 . . . . . . 7 ((𝑧𝐵𝜓) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → (∃𝑥𝐴 𝑥 = 𝐷 → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))))
3130com23 86 . . . . . 6 ((𝑧𝐵𝜓) → (∃𝑥𝐴 𝑥 = 𝐷 → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))))
329, 31syl5bi 233 . . . . 5 ((𝑧𝐵𝜓) → (𝐷𝐴 → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))))
3332expimpd 445 . . . 4 (𝑧𝐵 → ((𝜓𝐷𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))))
348, 33rexlimi 3171 . . 3 (∃𝑧𝐵 (𝜓𝐷𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
355, 34sylbi 208 . 2 (∃𝑦𝐵 (𝜑𝐶𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
361, 2reusv3i 5041 . 2 (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
3735, 36impbid1 216 1 (∃𝑦𝐵 (𝜑𝐶𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3737  df-nul 4082
This theorem is referenced by:  cdleme25b  36313  cdleme29b  36334  cdlemk28-3  36867  dihlsscpre  37193  mapdh9a  37748  mapdh9aOLDN  37749
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