Theorem bj-axc14nf 34685

Description: Proof of a version of axc14 2464 using the "nonfree" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axc14nf	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦))

Proof of Theorem bj-axc14nf

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-nfeel2 34684	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑥 ∈ 𝑡)
2		elequ2 2129	. 2 ⊢ (𝑡 = 𝑦 → (𝑥 ∈ 𝑡 ↔ 𝑥 ∈ 𝑦))
3	1, 2	bj-dvelimdv1 34682	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-13 2373

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791

This theorem is referenced by:  bj-axc14  34686