Theorem vtocl2 2776

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 2729	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝐴
3		vtocl2.2	. . . . . 6 ⊢ 𝐵 ∈ V
4	3	isseti 2729	. . . . 5 ⊢ ∃𝑦 𝑦 = 𝐵
5		eeanv 1919	. . . . . 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 143	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 → 𝜓))
8	7	2eximi 1588	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓))
9	5, 8	sylbir 134	. . . . 5 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓))
10	2, 4, 9	mp2an 423	. . . 4 ⊢ ∃𝑥∃𝑦(𝜑 → 𝜓)
11		nfv 1515	. . . . 5 ⊢ Ⅎ𝑦𝜓
12	11	19.36-1 1660	. . . 4 ⊢ (∃𝑦(𝜑 → 𝜓) → (∀𝑦𝜑 → 𝜓))
13	10, 12	eximii 1589	. . 3 ⊢ ∃𝑥(∀𝑦𝜑 → 𝜓)
14	13	19.36aiv 1888	. 2 ⊢ (∀𝑥∀𝑦𝜑 → 𝜓)
15		vtocl2.4	. . 3 ⊢ 𝜑
16	15	ax-gen 1436	. 2 ⊢ ∀𝑦𝜑
17	14, 16	mpg 1438	1 ⊢ 𝜓

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1340   = wceq 1342  ∃wex 1479   ∈ wcel 2135  Vcvv 2721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-v 2723

This theorem is referenced by:  undifexmid  4166  caovord  6004  ctssexmid  7105