Theorem eqvincg 3610

Description: A variable introduction law for class equality, closed form. (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 2847	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		ax-1 6	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 = 𝐵 → 𝑥 = 𝐴))
3		eqtr 2785	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝐴 = 𝐵) → 𝑥 = 𝐵)
4	3	ex 417	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 = 𝐵 → 𝑥 = 𝐵))
5	2, 4	jca 520	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)))
6	5	eximi 1858	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)))
7		pm3.43 478	. . . . 5 ⊢ (((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)) → (𝐴 = 𝐵 → (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
8	7	eximi 1858	. . . 4 ⊢ (∃𝑥((𝐴 = 𝐵 → 𝑥 = 𝐴) ∧ (𝐴 = 𝐵 → 𝑥 = 𝐵)) → ∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
9	1, 6, 8	3syl 19	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
10		19.37v 2020	. . 3 ⊢ (∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) ↔ (𝐴 = 𝐵 → ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
11	9, 10	sylib 221	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
12		eqtr2 2786	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → 𝐴 = 𝐵)
13	12	exlimiv 1953	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → 𝐴 = 𝐵)
14	11, 13	impbid1 228	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840

This theorem is referenced by:  eqvinc  3611  funcnv5mpt  32924