MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reusv3i Structured version   Visualization version   GIF version

Theorem reusv3i 5074
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 2809 . . . . . 6 (𝑦 = 𝑧 → (𝑥 = 𝐶𝑥 = 𝐷))
41, 3imbi12d 336 . . . . 5 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝐶) ↔ (𝜓𝑥 = 𝐷)))
54cbvralv 3354 . . . 4 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∀𝑧𝐵 (𝜓𝑥 = 𝐷))
65biimpi 208 . . 3 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑧𝐵 (𝜓𝑥 = 𝐷))
7 raaanv 4274 . . . 4 (∀𝑦𝐵𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) ↔ (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∀𝑧𝐵 (𝜓𝑥 = 𝐷)))
8 prth 844 . . . . . 6 (((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ((𝜑𝜓) → (𝑥 = 𝐶𝑥 = 𝐷)))
9 eqtr2 2819 . . . . . 6 ((𝑥 = 𝐶𝑥 = 𝐷) → 𝐶 = 𝐷)
108, 9syl6 35 . . . . 5 (((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ((𝜑𝜓) → 𝐶 = 𝐷))
11102ralimi 3134 . . . 4 (∀𝑦𝐵𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
127, 11sylbir 227 . . 3 ((∀𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∀𝑧𝐵 (𝜓𝑥 = 𝐷)) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
136, 12mpdan 679 . 2 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
1413rexlimivw 3210 1 (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wral 3089  wrex 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-v 3387  df-dif 3772  df-nul 4116
This theorem is referenced by:  reusv3  5075
  Copyright terms: Public domain W3C validator