Theorem bj-eunex 33375

Description: Remove dependency on ax-13 2334 from eunex 5101. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-eunex	⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Proof of Theorem bj-eunex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-dtru 33373	. . . 4 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2		albi 1862	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥 𝑥 = 𝑦))
3	1, 2	mtbiri 319	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
4	3	exlimiv 1973	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
5		eu6 2592	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6		exnal 1870	. 2 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
7	4, 5, 6	3imtr4i 284	1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198  ∀wal 1599  ∃wex 1823  ∃!weu 2586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-nul 5025  ax-pow 5077

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-mo 2551  df-eu 2587

This theorem is referenced by: (None)