Theorem axc16nf 2298

Description: If dtru 5405 is false, then there is only one element in the universe, so everything satisfies Ⅎ. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2194. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2178. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
axc16nf	⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)

Proof of Theorem axc16nf

Step	Hyp	Ref	Expression
1		axc16g 2295	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑧 ¬ 𝜑))
2		eximal 1815	. . . 4 ⊢ ((∃𝑧𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑧 ¬ 𝜑))
3	1, 2	sylibr 237	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → 𝜑))
4		axc16g 2295	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
5	3, 4	syld 48	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → ∀𝑧𝜑))
6	5	nfd 1823	1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  nfsbd  2552  exists2  2687