Theorem dveel2ALT 2191

Description: Version of dveel2 2020 using ax-16 2144 instead of ax-17 1616. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dveel2ALT	⊢ (¬ ∀x x = y → (z ∈ y → ∀x z ∈ y))

Distinct variable group:   x,z

Proof of Theorem dveel2ALT

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax17el 2189	. 2 ⊢ (z ∈ w → ∀x z ∈ w)
2		ax17el 2189	. 2 ⊢ (z ∈ y → ∀w z ∈ y)
3		elequ2 1715	. 2 ⊢ (w = y → (z ∈ w ↔ z ∈ y))
4	1, 2, 3	dvelimh 1964	1 ⊢ (¬ ∀x x = y → (z ∈ y → ∀x z ∈ y))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-15 2143  ax-16 2144

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)