Theorem vtocl3ga 2834

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 2339	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2339	. 2 ⊢ Ⅎ𝑦𝐴
3		nfcv 2339	. 2 ⊢ Ⅎ𝑧𝐴
4		nfcv 2339	. 2 ⊢ Ⅎ𝑦𝐵
5		nfcv 2339	. 2 ⊢ Ⅎ𝑧𝐵
6		nfcv 2339	. 2 ⊢ Ⅎ𝑧𝐶
7		nfv 1542	. 2 ⊢ Ⅎ𝑥𝜓
8		nfv 1542	. 2 ⊢ Ⅎ𝑦𝜒
9		nfv 1542	. 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 2833	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  preq12bg  3803  pocl  4338  sowlin  4355