![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3590 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵)) |
3 | eqvincf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfeq2 2972 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
5 | eqvincf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | nfeq2 2972 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵 |
7 | 4, 6 | nfan 1900 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) |
8 | nfv 1915 | . . 3 ⊢ Ⅎ𝑦(𝑥 = 𝐴 ∧ 𝑥 = 𝐵) | |
9 | eqeq1 2802 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴)) | |
10 | eqeq1 2802 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐵 ↔ 𝑥 = 𝐵)) | |
11 | 9, 10 | anbi12d 633 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝐵))) |
12 | 7, 8, 11 | cbvexv1 2351 | . 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
13 | 2, 12 | bitri 278 | 1 ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Ⅎwnfc 2936 Vcvv 3441 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-cleq 2791 df-clel 2870 df-nfc 2938 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |