Theorem exsbim 2021

Description: One direction of the equivalence in exsb 2389 is based on fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.)

Assertion

Ref	Expression
exsbim	⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exsbim

Step	Hyp	Ref	Expression
1		alequexv 2020	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
2	1	exlimiv 1949	1 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∀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

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

This theorem is referenced by:  spsbe  2114  eu6  2600  eu6im  2601  eu6w  43218