Theorem bj-axc14 37376

Description: Alternate proof of axc14 2501 (even when inlining the above results, this gives a shorter proof). (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axc14	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))

Proof of Theorem bj-axc14

Step	Hyp	Ref	Expression
1		bj-axc14nf 37375	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦))
2		nf5r 2236	. . 3 ⊢ (Ⅎ𝑧 𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))
3	2	a1i 11	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (Ⅎ𝑧 𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
4	1, 3	syld 48	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1565  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-13 2410

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811

This theorem is referenced by: (None)