Theorem nd1 10475

Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2372. (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 9483	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
2		stdpc4 2071	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑧 → [𝑧 / 𝑦]𝑦 ∈ 𝑧)
3	1	nfnth 1803	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧
4		elequ1 2118	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑧 ↔ 𝑧 ∈ 𝑧))
5	3, 4	sbie 2502	. . . 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 2430	. 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 1539  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-13 2372  ax-sep 5234  ax-pr 5370  ax-reg 9478

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-sb 2068

This theorem is referenced by:  axrepnd  10482  axinfndlem1  10493  axinfnd  10494  axacndlem1  10495  axacndlem2  10496