Theorem vtocl2 2856

Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
vtocl2.1	⊢ 𝐴 ∈ V
vtocl2.2	⊢ 𝐵 ∈ V
vtocl2.3	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
vtocl2.4	⊢ 𝜑

Assertion

Ref	Expression
vtocl2	⊢ 𝜓

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem vtocl2

Step	Hyp	Ref	Expression
1		vtocl2.1	. . . . . 6 ⊢ 𝐴 ∈ V
2	1	isseti 2808	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝐴
3		vtocl2.2	. . . . . 6 ⊢ 𝐵 ∈ V
4	3	isseti 2808	. . . . 5 ⊢ ∃𝑦 𝑦 = 𝐵
5		eeanv 1983	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6		vtocl2.3	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
7	6	biimpd 144	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 → 𝜓))
8	7	2eximi 1647	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓))
9	5, 8	sylbir 135	. . . . 5 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓))
10	2, 4, 9	mp2an 426	. . . 4 ⊢ ∃𝑥∃𝑦(𝜑 → 𝜓)
11		nfv 1574	. . . . 5 ⊢ Ⅎ𝑦𝜓
12	11	19.36-1 1719	. . . 4 ⊢ (∃𝑦(𝜑 → 𝜓) → (∀𝑦𝜑 → 𝜓))
13	10, 12	eximii 1648	. . 3 ⊢ ∃𝑥(∀𝑦𝜑 → 𝜓)
14	13	19.36aiv 1948	. 2 ⊢ (∀𝑥∀𝑦𝜑 → 𝜓)
15		vtocl2.4	. . 3 ⊢ 𝜑
16	15	ax-gen 1495	. 2 ⊢ ∀𝑦𝜑
17	14, 16	mpg 1497	1 ⊢ 𝜓

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1393   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801

This theorem is referenced by:  undifexmid  4276  caovord  6176  ctssexmid  7313