Theorem eqvincf 2855

Description: A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)

Hypotheses

Ref	Expression
eqvincf.1	⊢ Ⅎ𝑥𝐴
eqvincf.2	⊢ Ⅎ𝑥𝐵
eqvincf.3	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
eqvincf	⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))

Proof of Theorem eqvincf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqvincf.3	. . 3 ⊢ 𝐴 ∈ V
2	1	eqvinc 2853	. 2 ⊢ (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵))
3		eqvincf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4	3	nfeq2 2324	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴
5		eqvincf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	5	nfeq2 2324	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵
7	4, 6	nfan 1558	. . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 = 𝐵)
8		nfv 1521	. . 3 ⊢ Ⅎ𝑦(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)
9		eqeq1 2177	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴))
10		eqeq1 2177	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐵 ↔ 𝑥 = 𝐵))
11	9, 10	anbi12d 470	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
12	7, 8, 11	cbvex 1749	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))
13	2, 12	bitri 183	1 ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Ⅎwnfc 2299  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by: (None)