Theorem dveel2 2146

Description: Quantifier introduction when one pair of variables is disjoint. (Contributed by NM, 2-Jan-2002.)

Assertion

Ref	Expression
dveel2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))

Distinct variable group:   𝑥,𝑧

Proof of Theorem dveel2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1514	. 2 ⊢ (𝑧 ∈ 𝑤 → ∀𝑥 𝑧 ∈ 𝑤)
2		ax-17 1514	. 2 ⊢ (𝑧 ∈ 𝑦 → ∀𝑤 𝑧 ∈ 𝑦)
3		elequ2 2141	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
4	1, 2, 3	dvelimf 2003	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1341

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-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by: (None)