Theorem ax12ev2 2218

Description: Version of ax12v2 2217 rewritten to use an existential quantifier. One direction of sbalex 2280 without the universal quantifier, avoiding ax-10 2178. (Contributed by SN, 14-Aug-2025.)

Assertion

Ref	Expression
ax12ev2	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12ev2

Step	Hyp	Ref	Expression
1		exnalimn 1867	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
2		ax12v2 2217	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
3	2	con1d 146	. . 3 ⊢ (𝑥 = 𝑦 → (¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) → 𝜑))
4	1, 3	biimtrid 245	. 2 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))
5	4	com12 33	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1561  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  sbalex  2280  mopick  2655  sbalexi  42842