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Theorem reusv3i 5295
Description: Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
Hypotheses
Ref Expression
reusv3.1 (𝑦 = 𝑧 → (𝜑𝜓))
reusv3.2 (𝑦 = 𝑧𝐶 = 𝐷)
Assertion
Ref Expression
reusv3i (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐶,𝑧   𝑥,𝐷,𝑦   𝜑,𝑥,𝑧   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑧)   𝐴(𝑥,𝑦,𝑧)   𝐶(𝑦)   𝐷(𝑧)

Proof of Theorem reusv3i
StepHypRef Expression
1 reusv3.1 . . . . . 6 (𝑦 = 𝑧 → (𝜑𝜓))
2 reusv3.2 . . . . . . 7 (𝑦 = 𝑧𝐶 = 𝐷)
32eqeq2d 2829 . . . . . 6 (𝑦 = 𝑧 → (𝑥 = 𝐶𝑥 = 𝐷))
41, 3imbi12d 346 . . . . 5 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝐶) ↔ (𝜓𝑥 = 𝐷)))
54cbvralvw 3447 . . . 4 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∀𝑧𝐵 (𝜓𝑥 = 𝐷))
65biimpi 217 . . 3 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑧𝐵 (𝜓𝑥 = 𝐷))
7 raaanv 4457 . . . 4 (∀𝑦𝐵𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) ↔ (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∀𝑧𝐵 (𝜓𝑥 = 𝐷)))
8 anim12 805 . . . . . 6 (((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ((𝜑𝜓) → (𝑥 = 𝐶𝑥 = 𝐷)))
9 eqtr2 2839 . . . . . 6 ((𝑥 = 𝐶𝑥 = 𝐷) → 𝐶 = 𝐷)
108, 9syl6 35 . . . . 5 (((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ((𝜑𝜓) → 𝐶 = 𝐷))
11102ralimi 3158 . . . 4 (∀𝑦𝐵𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
127, 11sylbir 236 . . 3 ((∀𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∀𝑧𝐵 (𝜓𝑥 = 𝐷)) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
136, 12mpdan 683 . 2 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
1413rexlimivw 3279 1 (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wral 3135  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-dif 3936  df-nul 4289
This theorem is referenced by:  reusv3  5296
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