Theorem vtocl3ga 3496

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)

Hypotheses

Ref	Expression
vtocl3ga.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl3ga.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl3ga.3	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
vtocl3ga.4	⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑)

Assertion

Ref	Expression
vtocl3ga	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝐷,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜓,𝑥   𝜒,𝑦   𝜃,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧)   𝜃(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem vtocl3ga

Step	Hyp	Ref	Expression
1		nfcv 2919	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2919	. 2 ⊢ Ⅎ𝑦𝐴
3		nfcv 2919	. 2 ⊢ Ⅎ𝑧𝐴
4		nfcv 2919	. 2 ⊢ Ⅎ𝑦𝐵
5		nfcv 2919	. 2 ⊢ Ⅎ𝑧𝐵
6		nfcv 2919	. 2 ⊢ Ⅎ𝑧𝐶
7		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜓
8		nfv 1915	. 2 ⊢ Ⅎ𝑦𝜒
9		nfv 1915	. 2 ⊢ Ⅎ𝑧𝜃
10		vtocl3ga.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
11		vtocl3ga.2	. 2 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
12		vtocl3ga.3	. 2 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
13		vtocl3ga.4	. 2 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑)
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	vtocl3gaf 3495	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-v 3411

This theorem is referenced by:  preq12bg  4741  pocl  5450  jensenlem2  25672  wrdt2ind  30749  xpord3pred  33353