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Theorem iotavalsb 38113
 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 2049 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 df-eu 2473 . . 3 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 iotavalb 38110 . . . 4 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
4 dfsbcq 3419 . . . . 5 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
54eqcoms 2629 . . . 4 ((℩𝑥𝜑) = 𝑦 → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
63, 5syl6bi 243 . . 3 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓)))
72, 6sylbir 225 . 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 196  ∀wal 1478   = wceq 1480  ∃wex 1701  ∃!weu 2469  [wsbc 3417  ℩cio 5808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-v 3188  df-sbc 3418  df-un 3560  df-sn 4149  df-pr 4151  df-uni 4403  df-iota 5810 This theorem is referenced by: (None)
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