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Theorem reusv3i 4375
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 2149 . . . . . 6 (𝑦 = 𝑧 → (𝑥 = 𝐶𝑥 = 𝐷))
41, 3imbi12d 233 . . . . 5 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝐶) ↔ (𝜓𝑥 = 𝐷)))
54cbvralv 2652 . . . 4 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∀𝑧𝐵 (𝜓𝑥 = 𝐷))
65biimpi 119 . . 3 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑧𝐵 (𝜓𝑥 = 𝐷))
7 raaanv 3465 . . . 4 (∀𝑦𝐵𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) ↔ (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∀𝑧𝐵 (𝜓𝑥 = 𝐷)))
8 anim12 341 . . . . . . 7 (((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ((𝜑𝜓) → (𝑥 = 𝐶𝑥 = 𝐷)))
9 eqtr2 2156 . . . . . . 7 ((𝑥 = 𝐶𝑥 = 𝐷) → 𝐶 = 𝐷)
108, 9syl6 33 . . . . . 6 (((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ((𝜑𝜓) → 𝐶 = 𝐷))
1110ralimi 2493 . . . . 5 (∀𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ∀𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
1211ralimi 2493 . . . 4 (∀𝑦𝐵𝑧𝐵 ((𝜑𝑥 = 𝐶) ∧ (𝜓𝑥 = 𝐷)) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
137, 12sylbir 134 . . 3 ((∀𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∀𝑧𝐵 (𝜓𝑥 = 𝐷)) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
146, 13mpdan 417 . 2 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
1514rexlimivw 2543 1 (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wral 2414  wrex 2415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420
This theorem is referenced by:  reusv3  4376
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