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Theorem iotavalsb 42805
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 2175 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 eu6 2569 . . 3 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 iotavalb 42802 . . . 4 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
4 dfsbcq 3745 . . . . 5 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
54eqcoms 2741 . . . 4 ((℩𝑥𝜑) = 𝑦 → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
63, 5syl6bi 253 . . 3 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓)))
72, 6sylbir 234 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓)))
81, 7mpcom 38 1 (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540   = wceq 1542  wex 1782  ∃!weu 2563  [wsbc 3743  cio 6450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-sbc 3744  df-un 3919  df-in 3921  df-ss 3931  df-sn 4591  df-pr 4593  df-uni 4870  df-iota 6452
This theorem is referenced by: (None)
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