Theorem eqvincf 3611

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 3610	. 2 ⊢ (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵))
3		eqvincf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4	3	nfeq2 2944	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴
5		eqvincf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	5	nfeq2 2944	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵
7	4, 6	nfan 1932	. . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 = 𝐵)
8		nfv 1947	. . 3 ⊢ Ⅎ𝑦(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)
9		eqeq1 2769	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴))
10		eqeq1 2769	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐵 ↔ 𝑥 = 𝐵))
11	9, 10	anbi12d 644	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
12	7, 8, 11	cbvexv1 2376	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))
13	2, 12	bitri 278	1 ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Ⅎwnfc 2912  Vcvv 3457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914

This theorem is used by: (None)