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Mirrors > Home > MPE Home > Th. List > eqvincf | Structured version Visualization version GIF version |
Description: A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.) |
Ref | Expression |
---|---|
eqvincf.1 | ⊢ Ⅎ𝑥𝐴 |
eqvincf.2 | ⊢ Ⅎ𝑥𝐵 |
eqvincf.3 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eqvincf | ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvincf.3 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | eqvinc 3641 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵)) |
3 | eqvincf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfeq2 2995 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
5 | eqvincf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | nfeq2 2995 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵 |
7 | 4, 6 | nfan 1896 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) |
8 | nfv 1911 | . . 3 ⊢ Ⅎ𝑦(𝑥 = 𝐴 ∧ 𝑥 = 𝐵) | |
9 | eqeq1 2825 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴)) | |
10 | eqeq1 2825 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐵 ↔ 𝑥 = 𝐵)) | |
11 | 9, 10 | anbi12d 632 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝐵))) |
12 | 7, 8, 11 | cbvexv1 2358 | . 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
13 | 2, 12 | bitri 277 | 1 ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Ⅎwnfc 2961 Vcvv 3494 |
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-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-cleq 2814 df-clel 2893 df-nfc 2963 |
This theorem is referenced by: (None) |
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