Theorem dveel2ALT 37804

Description: Alternate proof of dveel2 2461 using ax-c16 37757 instead of ax-5 1913. (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		ax5el 37802	. 2 ⊢ (𝑧 ∈ 𝑤 → ∀𝑥 𝑧 ∈ 𝑤)
2		ax5el 37802	. 2 ⊢ (𝑧 ∈ 𝑦 → ∀𝑤 𝑧 ∈ 𝑦)
3		elequ2 2121	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
4	1, 2, 3	dvelimh 2449	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371  ax-c14 37756  ax-c16 37757

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)