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Theorem sbc2or 3732
Description: The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for [𝐴 / 𝑥]𝜑 behavior at proper classes, matching the sbc5 3751 (false) and sbc6 3753 (true) conclusions. This is interesting since dfsbcq 3725 and dfsbcq2 3726 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem does not tell us that it is always one or the other at proper classes; it could "flip" between false (the first disjunct) and true (the second disjunct) as a function of some other variable 𝑦 that 𝜑 or 𝐴 may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbc2or (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sbc2or
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3726 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 eqeq2 2813 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
32anbi1d 632 . . . . 5 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
43exbidv 1922 . . . 4 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
5 sb5 2275 . . . 4 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
61, 4, 5vtoclbg 3520 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
76orcd 870 . 2 (𝐴 ∈ V → (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑))))
8 pm5.15 1010 . . 3 (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝐴𝜑)))
9 vex 3447 . . . . . . . . . 10 𝑥 ∈ V
10 eleq1 2880 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
119, 10mpbii 236 . . . . . . . . 9 (𝑥 = 𝐴𝐴 ∈ V)
1211adantr 484 . . . . . . . 8 ((𝑥 = 𝐴𝜑) → 𝐴 ∈ V)
1312con3i 157 . . . . . . 7 𝐴 ∈ V → ¬ (𝑥 = 𝐴𝜑))
1413nexdv 1937 . . . . . 6 𝐴 ∈ V → ¬ ∃𝑥(𝑥 = 𝐴𝜑))
1511con3i 157 . . . . . . . 8 𝐴 ∈ V → ¬ 𝑥 = 𝐴)
1615pm2.21d 121 . . . . . . 7 𝐴 ∈ V → (𝑥 = 𝐴𝜑))
1716alrimiv 1928 . . . . . 6 𝐴 ∈ V → ∀𝑥(𝑥 = 𝐴𝜑))
1814, 172thd 268 . . . . 5 𝐴 ∈ V → (¬ ∃𝑥(𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
1918bibi2d 346 . . . 4 𝐴 ∈ V → (([𝐴 / 𝑥]𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝐴𝜑)) ↔ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑))))
2019orbi2d 913 . . 3 𝐴 ∈ V → ((([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝐴𝜑))) ↔ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))))
218, 20mpbii 236 . 2 𝐴 ∈ V → (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑))))
227, 21pm2.61i 185 1 (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  wal 1536   = wceq 1538  wex 1781  [wsb 2069  wcel 2112  Vcvv 3444  [wsbc 3723
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 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-sbc 3724
This theorem is referenced by: (None)
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