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Mirrors > Home > MPE Home > Th. List > sbeqalb | Structured version Visualization version GIF version |
Description: Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) |
Ref | Expression |
---|---|
sbeqalb | ⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bibi1 351 | . . . . 5 ⊢ ((𝜑 ↔ 𝑥 = 𝐴) → ((𝜑 ↔ 𝑥 = 𝐵) ↔ (𝑥 = 𝐴 ↔ 𝑥 = 𝐵))) | |
2 | 1 | biimpa 476 | . . . 4 ⊢ (((𝜑 ↔ 𝑥 = 𝐴) ∧ (𝜑 ↔ 𝑥 = 𝐵)) → (𝑥 = 𝐴 ↔ 𝑥 = 𝐵)) |
3 | 2 | biimpd 228 | . . 3 ⊢ (((𝜑 ↔ 𝑥 = 𝐴) ∧ (𝜑 ↔ 𝑥 = 𝐵)) → (𝑥 = 𝐴 → 𝑥 = 𝐵)) |
4 | 3 | alanimi 1820 | . 2 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵)) |
5 | sbceqal 3778 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵)) | |
6 | 4, 5 | syl5 34 | 1 ⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1542 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-v 3424 |
This theorem is referenced by: iotaval 6392 |
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