Theorem sbc2or 3765

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 3784 (false) and sbc6 3787 (true) conclusions. This is interesting since dfsbcq 3758 and dfsbcq2 3759 (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 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sbc2or

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3759	. . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2		eqeq2 2742	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
3	2	anbi1d 631	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑)))
4	3	exbidv 1921	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
5		sb5 2276	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
6	1, 4, 5	vtoclbg 3526	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
7	6	orcd 873	. 2 ⊢ (𝐴 ∈ V → (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))))
8		pm5.15 1014	. . 3 ⊢ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
9		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
10		eleq1 2817	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
11	9, 10	mpbii 233	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝐴 ∈ V)
13	12	con3i 154	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ¬ (𝑥 = 𝐴 ∧ 𝜑))
14	13	nexdv 1936	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
15	11	con3i 154	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → ¬ 𝑥 = 𝐴)
16	15	pm2.21d 121	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝑥 = 𝐴 → 𝜑))
17	16	alrimiv 1927	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ∀𝑥(𝑥 = 𝐴 → 𝜑))
18	14, 17	2thd 265	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
19	18	bibi2d 342	. . . 4 ⊢ (¬ 𝐴 ∈ V → (([𝐴 / 𝑥]𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ↔ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))))
20	19	orbi2d 915	. . 3 ⊢ (¬ 𝐴 ∈ V → ((([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) ↔ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))))
21	8, 20	mpbii 233	. 2 ⊢ (¬ 𝐴 ∈ V → (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))))
22	7, 21	pm2.61i 182	1 ⊢ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  ∀wal 1538   = wceq 1540  ∃wex 1779  [wsb 2065   ∈ wcel 2109  Vcvv 3450  [wsbc 3756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-sbc 3757

This theorem is referenced by: (None)