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Theorem iotavalsb 37656
Description: Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotavalsb (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem iotavalsb
StepHypRef Expression
1 19.8a 2039 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 df-eu 2462 . . 3 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 iotavalb 37653 . . . 4 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
4 dfsbcq 3404 . . . . 5 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
54eqcoms 2618 . . . 4 ((℩𝑥𝜑) = 𝑦 → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
63, 5syl6bi 242 . . 3 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓)))
72, 6sylbir 224 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓)))
81, 7mpcom 37 1 (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473   = wceq 1475  wex 1695  ∃!weu 2458  [wsbc 3402  cio 5766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-sbc 3403  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768
This theorem is referenced by: (None)
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