Theorem dveel2 2185

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 1548	. 2 ⊢ (𝑧 ∈ 𝑤 → ∀𝑥 𝑧 ∈ 𝑤)
2		ax-17 1548	. 2 ⊢ (𝑧 ∈ 𝑦 → ∀𝑤 𝑧 ∈ 𝑦)
3		elequ2 2180	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
4	1, 2, 3	dvelimf 2042	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1370

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785

This theorem is referenced by: (None)