Theorem dveel2 2020

Description: Quantifier introduction when one pair of variables is distinct. (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		elequ2 1715	. 2 ⊢ (w = y → (z ∈ w ↔ z ∈ y))
2	1	dvelimv 1939	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

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:  ax15  2021