Theorem nd1 10606

Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.)

Assertion

Ref	Expression
nd1	⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧)

Proof of Theorem nd1

Step	Hyp	Ref	Expression
1		elirrv 9615	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
2		stdpc4 2069	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑧 → [𝑧 / 𝑦]𝑦 ∈ 𝑧)
3	1	nfnth 1802	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧
4		elequ1 2116	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑧 ↔ 𝑧 ∈ 𝑧))
5	3, 4	sbie 2507	. . . 4 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝑧 ↔ 𝑧 ∈ 𝑧)
6	2, 5	sylib 218	. . 3 ⊢ (∀𝑦 𝑦 ∈ 𝑧 → 𝑧 ∈ 𝑧)
7	1, 6	mto 197	. 2 ⊢ ¬ ∀𝑦 𝑦 ∈ 𝑧
8		axc11 2435	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦 ∈ 𝑧 → ∀𝑦 𝑦 ∈ 𝑧))
9	7, 8	mtoi 199	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-13 2377  ax-ext 2708  ax-sep 5271  ax-pr 5407  ax-reg 9611

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-v 3466  df-un 3936  df-sn 4607  df-pr 4609

This theorem is referenced by:  axrepnd  10613  axinfndlem1  10624  axinfnd  10625  axacndlem1  10626  axacndlem2  10627