Theorem ax12ev2 2214

Description: Version of ax12v2 2213 rewritten to use an existential quantifier. One direction of sbalex 2276 without the universal quantifier, avoiding ax-10 2174. (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 1863	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
2		ax12v2 2213	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
3	2	con1d 145	. . 3 ⊢ (𝑥 = 𝑦 → (¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) → 𝜑))
4	1, 3	biimtrid 244	. 2 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))
5	4	com12 32	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799

This theorem is referenced by:  sbalex  2276  mopick  2651  sbalexi  42794