Theorem dveel2ALT 39713

Description: Alternate proof of dveel2 2494 using ax-c16 39666 instead of ax-5 1940. (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 39711	. 2 ⊢ (𝑧 ∈ 𝑤 → ∀𝑥 𝑧 ∈ 𝑤)
2		ax5el 39711	. 2 ⊢ (𝑧 ∈ 𝑦 → ∀𝑤 𝑧 ∈ 𝑦)
3		elequ2 2158	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
4	1, 2, 3	dvelimh 2482	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-c14 39665  ax-c16 39666

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814

This theorem is referenced by: (None)