Theorem axc16nf 2137

Description: If dtru 4857 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 2034. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2019. (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		df-ex 1705	. . . 4 ⊢ (∃𝑧𝜑 ↔ ¬ ∀𝑧 ¬ 𝜑)
2		axc16g 2134	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑧 ¬ 𝜑))
3	2	con1d 139	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑧 ¬ 𝜑 → 𝜑))
4	1, 3	syl5bi 232	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → 𝜑))
5		axc16g 2134	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
6	4, 5	syld 47	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → ∀𝑧𝜑))
7	6	nfd 1716	1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  nfsb  2440  nfsbd  2442