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Mirrors > Home > MPE Home > Th. List > sbc2or | Structured version Visualization version GIF version |
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 3801 (false) and sbc6 3805 (true) conclusions. This is interesting since dfsbcq 3775 and dfsbcq2 3776 (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.) |
Ref | Expression |
---|---|
sbc2or | ⊢ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3776 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | eqeq2 2737 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
3 | 2 | anbi1d 629 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑))) |
4 | 3 | exbidv 1916 | . . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
5 | sb5 2262 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
6 | 1, 4, 5 | vtoclbg 3535 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
7 | 6 | orcd 871 | . 2 ⊢ (𝐴 ∈ V → (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))) |
8 | pm5.15 1010 | . . 3 ⊢ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) | |
9 | vex 3465 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
10 | eleq1 2813 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V)) | |
11 | 9, 10 | mpbii 232 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) |
12 | 11 | adantr 479 | . . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝐴 ∈ V) |
13 | 12 | con3i 154 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ¬ (𝑥 = 𝐴 ∧ 𝜑)) |
14 | 13 | nexdv 1931 | . . . . . 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 1922 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
18 | 14, 17 | 2thd 264 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) |
19 | 18 | bibi2d 341 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (([𝐴 / 𝑥]𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ↔ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))) |
20 | 19 | orbi2d 913 | . . 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 232 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∀wal 1531 = wceq 1533 ∃wex 1773 [wsb 2059 ∈ wcel 2098 Vcvv 3461 [wsbc 3773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-sbc 3774 |
This theorem is referenced by: (None) |
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