Theorem vtocl2 3251

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 3199	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝐴
3		vtocl2.2	. . . . . 6 ⊢ 𝐵 ∈ V
4	3	isseti 3199	. . . . 5 ⊢ ∃𝑦 𝑦 = 𝐵
5		eeanv 2181	. . . . . 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 219	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 → 𝜓))
8	7	2eximi 1760	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓))
9	5, 8	sylbir 225	. . . . 5 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓))
10	2, 4, 9	mp2an 707	. . . 4 ⊢ ∃𝑥∃𝑦(𝜑 → 𝜓)
11		19.36v 1901	. . . . 5 ⊢ (∃𝑦(𝜑 → 𝜓) ↔ (∀𝑦𝜑 → 𝜓))
12	11	exbii 1771	. . . 4 ⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ ∃𝑥(∀𝑦𝜑 → 𝜓))
13	10, 12	mpbi 220	. . 3 ⊢ ∃𝑥(∀𝑦𝜑 → 𝜓)
14	13	19.36iv 1902	. 2 ⊢ (∀𝑥∀𝑦𝜑 → 𝜓)
15		vtocl2.4	. . 3 ⊢ 𝜑
16	15	ax-gen 1719	. 2 ⊢ ∀𝑦𝜑
17	14, 16	mpg 1721	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1987  Vcvv 3190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-ext 2601

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3192

This theorem is referenced by:  caovord  6810  sornom  9059  wloglei  10520  ipodrsima  17105  mpfind  19476  mclsppslem  31241  monotoddzzfi  37026