Theorem sbc2or 3780

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 3799 (false) and sbc6 3801 (true) conclusions. This is interesting since dfsbcq 3773 and dfsbcq2 3774 (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 3774	. . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2		eqeq2 2833	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
3	2	anbi1d 631	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑)))
4	3	exbidv 1918	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
5		sb5 2272	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
6	1, 4, 5	vtoclbg 3568	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
7	6	orcd 869	. 2 ⊢ (𝐴 ∈ V → (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))))
8		pm5.15 1009	. . 3 ⊢ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
9		vex 3497	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
10		eleq1 2900	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
11	9, 10	mpbii 235	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
12	11	adantr 483	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝐴 ∈ V)
13	12	con3i 157	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ¬ (𝑥 = 𝐴 ∧ 𝜑))
14	13	nexdv 1933	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
15	11	con3i 157	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → ¬ 𝑥 = 𝐴)
16	15	pm2.21d 121	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝑥 = 𝐴 → 𝜑))
17	16	alrimiv 1924	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ∀𝑥(𝑥 = 𝐴 → 𝜑))
18	14, 17	2thd 267	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
19	18	bibi2d 345	. . . 4 ⊢ (¬ 𝐴 ∈ V → (([𝐴 / 𝑥]𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ↔ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))))
20	19	orbi2d 912	. . 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 235	. 2 ⊢ (¬ 𝐴 ∈ V → (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))))
22	7, 21	pm2.61i 184	1 ⊢ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843  ∀wal 1531   = wceq 1533  ∃wex 1776  [wsb 2065   ∈ wcel 2110  Vcvv 3494  [wsbc 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3496  df-sbc 3772

This theorem is referenced by: (None)