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Theorem reusv3 4906
 Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 4896 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 2715 . . . . 5 (𝑦 = 𝑧 → (𝐶𝐴𝐷𝐴))
41, 3anbi12d 747 . . . 4 (𝑦 = 𝑧 → ((𝜑𝐶𝐴) ↔ (𝜓𝐷𝐴)))
54cbvrexv 3202 . . 3 (∃𝑦𝐵 (𝜑𝐶𝐴) ↔ ∃𝑧𝐵 (𝜓𝐷𝐴))
6 nfra2 2975 . . . . 5 𝑧𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)
7 nfv 1883 . . . . 5 𝑧𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)
86, 7nfim 1865 . . . 4 𝑧(∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
9 risset 3091 . . . . . 6 (𝐷𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐷)
10 ralcom 3127 . . . . . . . . . . . . . 14 (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧𝐵𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
11 impexp 461 . . . . . . . . . . . . . . . . . 18 (((𝜑𝜓) → 𝐶 = 𝐷) ↔ (𝜑 → (𝜓𝐶 = 𝐷)))
12 bi2.04 375 . . . . . . . . . . . . . . . . . 18 ((𝜑 → (𝜓𝐶 = 𝐷)) ↔ (𝜓 → (𝜑𝐶 = 𝐷)))
1311, 12bitri 264 . . . . . . . . . . . . . . . . 17 (((𝜑𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → (𝜑𝐶 = 𝐷)))
1413ralbii 3009 . . . . . . . . . . . . . . . 16 (∀𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑦𝐵 (𝜓 → (𝜑𝐶 = 𝐷)))
15 r19.21v 2989 . . . . . . . . . . . . . . . 16 (∀𝑦𝐵 (𝜓 → (𝜑𝐶 = 𝐷)) ↔ (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
1614, 15bitri 264 . . . . . . . . . . . . . . 15 (∀𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
1716ralbii 3009 . . . . . . . . . . . . . 14 (∀𝑧𝐵𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧𝐵 (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
1810, 17bitri 264 . . . . . . . . . . . . 13 (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧𝐵 (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
19 rsp 2958 . . . . . . . . . . . . 13 (∀𝑧𝐵 (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)) → (𝑧𝐵 → (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))))
2018, 19sylbi 207 . . . . . . . . . . . 12 (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → (𝑧𝐵 → (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))))
2120com3l 89 . . . . . . . . . . 11 (𝑧𝐵 → (𝜓 → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))))
2221imp31 447 . . . . . . . . . 10 (((𝑧𝐵𝜓) ∧ ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)) → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))
23 eqeq1 2655 . . . . . . . . . . . . 13 (𝑥 = 𝐷 → (𝑥 = 𝐶𝐷 = 𝐶))
24 eqcom 2658 . . . . . . . . . . . . 13 (𝐷 = 𝐶𝐶 = 𝐷)
2523, 24syl6bb 276 . . . . . . . . . . . 12 (𝑥 = 𝐷 → (𝑥 = 𝐶𝐶 = 𝐷))
2625imbi2d 329 . . . . . . . . . . 11 (𝑥 = 𝐷 → ((𝜑𝑥 = 𝐶) ↔ (𝜑𝐶 = 𝐷)))
2726ralbidv 3015 . . . . . . . . . 10 (𝑥 = 𝐷 → (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
2822, 27syl5ibrcom 237 . . . . . . . . 9 (((𝑧𝐵𝜓) ∧ ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)) → (𝑥 = 𝐷 → ∀𝑦𝐵 (𝜑𝑥 = 𝐶)))
2928reximdv 3045 . . . . . . . 8 (((𝑧𝐵𝜓) ∧ ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)) → (∃𝑥𝐴 𝑥 = 𝐷 → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
3029ex 449 . . . . . . 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 232 . . . . 5 ((𝑧𝐵𝜓) → (𝐷𝐴 → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))))
3332expimpd 628 . . . 4 (𝑧𝐵 → ((𝜓𝐷𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))))
348, 33rexlimi 3053 . . 3 (∃𝑧𝐵 (𝜓𝐷𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
355, 34sylbi 207 . 2 (∃𝑦𝐵 (𝜑𝐶𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
361, 2reusv3i 4905 . 2 (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
3735, 36impbid1 215 1 (∃𝑦𝐵 (𝜑𝐶𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-nul 3949 This theorem is referenced by:  cdleme25b  35959  cdleme29b  35980  cdlemk28-3  36513  dihlsscpre  36840  mapdh9a  37396  mapdh9aOLDN  37397
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