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Theorem reusv3i 4845
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 2631 . . . . . 6 (𝑦 = 𝑧 → (𝑥 = 𝐶𝑥 = 𝐷))
41, 3imbi12d 334 . . . . 5 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝐶) ↔ (𝜓𝑥 = 𝐷)))
54cbvralv 3163 . . . 4 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∀𝑧𝐵 (𝜓𝑥 = 𝐷))
65biimpi 206 . . 3 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑧𝐵 (𝜓𝑥 = 𝐷))
7 raaanv 4061 . . . 4 (∀𝑦𝐵𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) ↔ (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∀𝑧𝐵 (𝜓𝑥 = 𝐷)))
8 prth 594 . . . . . 6 (((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ((𝜑𝜓) → (𝑥 = 𝐶𝑥 = 𝐷)))
9 eqtr2 2641 . . . . . 6 ((𝑥 = 𝐶𝑥 = 𝐷) → 𝐶 = 𝐷)
108, 9syl6 35 . . . . 5 (((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ((𝜑𝜓) → 𝐶 = 𝐷))
11102ralimi 2949 . . . 4 (∀𝑦𝐵𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
127, 11sylbir 225 . . 3 ((∀𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∀𝑧𝐵 (𝜓𝑥 = 𝐷)) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
136, 12mpdan 701 . 2 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
1413rexlimivw 3024 1 (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wral 2908  wrex 2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-v 3192  df-dif 3563  df-nul 3898
This theorem is referenced by:  reusv3  4846
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