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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 3637 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵)) |
3 | eqvincf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfeq2 2920 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
5 | eqvincf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | nfeq2 2920 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵 |
7 | 4, 6 | nfan 1902 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) |
8 | nfv 1917 | . . 3 ⊢ Ⅎ𝑦(𝑥 = 𝐴 ∧ 𝑥 = 𝐵) | |
9 | eqeq1 2736 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴)) | |
10 | eqeq1 2736 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐵 ↔ 𝑥 = 𝐵)) | |
11 | 9, 10 | anbi12d 631 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝐵))) |
12 | 7, 8, 11 | cbvexv1 2338 | . 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
13 | 2, 12 | bitri 274 | 1 ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Ⅎwnfc 2883 Vcvv 3474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 |
This theorem is referenced by: (None) |
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