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Theorem reusv3i 4191
 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 2051 . . . . . 6 (𝑦 = 𝑧 → (𝑥 = 𝐶𝑥 = 𝐷))
41, 3imbi12d 223 . . . . 5 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝐶) ↔ (𝜓𝑥 = 𝐷)))
54cbvralv 2533 . . . 4 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∀𝑧𝐵 (𝜓𝑥 = 𝐷))
65biimpi 113 . . 3 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑧𝐵 (𝜓𝑥 = 𝐷))
7 raaanv 3328 . . . 4 (∀𝑦𝐵𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) ↔ (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∀𝑧𝐵 (𝜓𝑥 = 𝐷)))
8 prth 326 . . . . . . 7 (((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ((𝜑𝜓) → (𝑥 = 𝐶𝑥 = 𝐷)))
9 eqtr2 2058 . . . . . . 7 ((𝑥 = 𝐶𝑥 = 𝐷) → 𝐶 = 𝐷)
108, 9syl6 29 . . . . . 6 (((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ((𝜑𝜓) → 𝐶 = 𝐷))
1110ralimi 2384 . . . . 5 (∀𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ∀𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
1211ralimi 2384 . . . 4 (∀𝑦𝐵𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
137, 12sylbir 125 . . 3 ((∀𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∀𝑧𝐵 (𝜓𝑥 = 𝐷)) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
146, 13mpdan 398 . 2 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
1514rexlimivw 2429 1 (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∀wral 2306  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312 This theorem is referenced by:  reusv3  4192
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