Theorem eqvinc 2853

Description: A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypothesis

Ref	Expression
eqvinc.1	⊢ 𝐴 ∈ V

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eqvinc

Step	Hyp	Ref	Expression
1		eqvinc.1	. . . . 5 ⊢ 𝐴 ∈ V
2	1	isseti 2738	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐴
3		ax-1 6	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 = 𝐵 → 𝑥 = 𝐴))
4		eqtr 2188	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝐴 = 𝐵) → 𝑥 = 𝐵)
5	4	ex 114	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 = 𝐵 → 𝑥 = 𝐵))
6	3, 5	jca 304	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)))
7	6	eximi 1593	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)))
8		pm3.43 597	. . . . 5 ⊢ (((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)) → (𝐴 = 𝐵 → (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
9	8	eximi 1593	. . . 4 ⊢ (∃𝑥((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)) → ∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
10	2, 7, 9	mp2b 8	. . 3 ⊢ ∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴 ∧ 𝑥 = 𝐵))
11	10	19.37aiv 1668	. 2 ⊢ (𝐴 = 𝐵 → ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))
12		eqtr2 2189	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → 𝐴 = 𝐵)
13	12	exlimiv 1591	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → 𝐴 = 𝐵)
14	11, 13	impbii 125	1 ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  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-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

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

This theorem is referenced by:  eqvincf  2855