Theorem axc16nf 2249

Description: If dtru 5442 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 2146. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2129. (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 2246	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑧 ¬ 𝜑))
2		eximal 1776	. . . 4 ⊢ ((∃𝑧𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑧 ¬ 𝜑))
3	1, 2	sylibr 233	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → 𝜑))
4		axc16g 2246	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
5	3, 4	syld 47	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → ∀𝑧𝜑))
6	5	nfd 1784	1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1531  ∃wex 1773  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-nf 1778

This theorem is referenced by:  nfsbd  2516  exists2  2653