Theorem eqvincg 2896

Description: A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)

Assertion

Ref	Expression
eqvincg	⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem eqvincg

Step	Hyp	Ref	Expression
1		elisset 2785	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		ax-1 6	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 = 𝐵 → 𝑥 = 𝐴))
3		eqtr 2222	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝐴 = 𝐵) → 𝑥 = 𝐵)
4	3	ex 115	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 = 𝐵 → 𝑥 = 𝐵))
5	2, 4	jca 306	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)))
6	5	eximi 1622	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)))
7		pm3.43 602	. . . . 5 ⊢ (((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)) → (𝐴 = 𝐵 → (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
8	7	eximi 1622	. . . 4 ⊢ (∃𝑥((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)) → ∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
9	1, 6, 8	3syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
10		nfv 1550	. . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐵
11	10	19.37-1 1696	. . 3 ⊢ (∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) → (𝐴 = 𝐵 → ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
12	9, 11	syl 14	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
13		eqtr2 2223	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → 𝐴 = 𝐵)
14	13	exlimiv 1620	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → 𝐴 = 𝐵)
15	12, 14	impbid1 142	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372  ∃wex 1514   ∈ wcel 2175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-v 2773

This theorem is referenced by:  dff13  5827  f1eqcocnv  5850