Theorem eqvincf 3552

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 3551	. 2 ⊢ (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵))
3		eqvincf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4	3	nfeq2 2941	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴
5		eqvincf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	5	nfeq2 2941	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵
7	4, 6	nfan 1862	. . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 = 𝐵)
8		nfv 1873	. . 3 ⊢ Ⅎ𝑦(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)
9		eqeq1 2776	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴))
10		eqeq1 2776	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐵 ↔ 𝑥 = 𝐵))
11	9, 10	anbi12d 621	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
12	7, 8, 11	cbvexv1 2278	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))
13	2, 12	bitri 267	1 ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))

Syntax hints:   ↔ wb 198   ∧ wa 387   = wceq 1507  ∃wex 1742   ∈ wcel 2050  Ⅎwnfc 2910  Vcvv 3409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-cleq 2765  df-clel 2840  df-nfc 2912

This theorem is referenced by: (None)