Theorem nd1 10604

Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2366. (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 9613	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
2		stdpc4 2064	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑧 → [𝑧 / 𝑦]𝑦 ∈ 𝑧)
3	1	nfnth 1797	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧
4		elequ1 2106	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑧 ↔ 𝑧 ∈ 𝑧))
5	3, 4	sbie 2496	. . . 4 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝑧 ↔ 𝑧 ∈ 𝑧)
6	2, 5	sylib 217	. . 3 ⊢ (∀𝑦 𝑦 ∈ 𝑧 → 𝑧 ∈ 𝑧)
7	1, 6	mto 196	. 2 ⊢ ¬ ∀𝑦 𝑦 ∈ 𝑧
8		axc11 2424	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦 ∈ 𝑧 → ∀𝑦 𝑦 ∈ 𝑧))
9	7, 8	mtoi 198	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1532  [wsb 2060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-13 2366  ax-ext 2698  ax-sep 5293  ax-pr 5423  ax-reg 9609

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-v 3471  df-un 3949  df-sn 4625  df-pr 4627

This theorem is referenced by:  axrepnd  10611  axinfndlem1  10622  axinfnd  10623  axacndlem1  10624  axacndlem2  10625