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Theorem sbc2or 3054
 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 3070 (false) and sbc6 3072 (true) conclusions. This is interesting since dfsbcq 3048 and dfsbcq2 3049 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem doesn't 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 / xφx(x = A φ)) ([̣A / xφx(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 3049 . . . 4 (y = A → ([y / x]φ ↔ [̣A / xφ))
2 eqeq2 2362 . . . . . 6 (y = A → (x = yx = A))
32anbi1d 685 . . . . 5 (y = A → ((x = y φ) ↔ (x = A φ)))
43exbidv 1626 . . . 4 (y = A → (x(x = y φ) ↔ x(x = A φ)))
5 sb5 2100 . . . 4 ([y / x]φx(x = y φ))
61, 4, 5vtoclbg 2915 . . 3 (A V → ([̣A / xφx(x = A φ)))
76orcd 381 . 2 (A V → (([̣A / xφx(x = A φ)) ([̣A / xφx(x = Aφ))))
8 pm5.15 859 . . 3 (([̣A / xφx(x = A φ)) ([̣A / xφ ↔ ¬ x(x = A φ)))
9 vex 2862 . . . . . . . . . 10 x V
10 eleq1 2413 . . . . . . . . . 10 (x = A → (x V ↔ A V))
119, 10mpbii 202 . . . . . . . . 9 (x = AA V)
1211adantr 451 . . . . . . . 8 ((x = A φ) → A V)
1312con3i 127 . . . . . . 7 A V → ¬ (x = A φ))
1413nexdv 1916 . . . . . 6 A V → ¬ x(x = A φ))
1511con3i 127 . . . . . . . 8 A V → ¬ x = A)
1615pm2.21d 98 . . . . . . 7 A V → (x = Aφ))
1716alrimiv 1631 . . . . . 6 A V → x(x = Aφ))
1814, 172thd 231 . . . . 5 A V → (¬ x(x = A φ) ↔ x(x = Aφ)))
1918bibi2d 309 . . . 4 A V → (([̣A / xφ ↔ ¬ x(x = A φ)) ↔ ([̣A / xφx(x = Aφ))))
2019orbi2d 682 . . 3 A V → ((([̣A / xφx(x = A φ)) ([̣A / xφ ↔ ¬ x(x = A φ))) ↔ (([̣A / xφx(x = A φ)) ([̣A / xφx(x = Aφ)))))
218, 20mpbii 202 . 2 A V → (([̣A / xφx(x = A φ)) ([̣A / xφx(x = Aφ))))
227, 21pm2.61i 156 1 (([̣A / xφx(x = A φ)) ([̣A / xφx(x = Aφ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  Vcvv 2859  [̣wsbc 3046 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047 This theorem is referenced by: (None)
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