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Theorem iotavalsb 41132
 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 2178 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 eu6 2634 . . 3 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 iotavalb 41129 . . . 4 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
4 dfsbcq 3722 . . . . 5 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
54eqcoms 2806 . . . 4 ((℩𝑥𝜑) = 𝑦 → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
63, 5syl6bi 256 . . 3 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓)))
72, 6sylbir 238 . 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 209  ∀wal 1536   = wceq 1538  ∃wex 1781  ∃!weu 2628  [wsbc 3720  ℩cio 6281 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-uni 4801  df-iota 6283 This theorem is referenced by: (None)
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