Theorem dveel1 2173

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

Assertion

Ref	Expression
dveel1	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → ∀𝑥 𝑦 ∈ 𝑧))

Distinct variable group:   𝑥,𝑧

Proof of Theorem dveel1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1537	. 2 ⊢ (𝑤 ∈ 𝑧 → ∀𝑥 𝑤 ∈ 𝑧)
2		ax-17 1537	. 2 ⊢ (𝑦 ∈ 𝑧 → ∀𝑤 𝑦 ∈ 𝑧)
3		elequ1 2168	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
4	1, 2, 3	dvelimf 2031	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → ∀𝑥 𝑦 ∈ 𝑧))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1362

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-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by: (None)