Theorem bj-alexbiex 36642

Description: Adding a second quantifier over the same variable is a transparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)

Assertion

Ref	Expression
bj-alexbiex	⊢ (∀𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)

Proof of Theorem bj-alexbiex

Step	Hyp	Ref	Expression
1		nfe1 2146	. 2 ⊢ Ⅎ𝑥∃𝑥𝜑
2	1	19.3 2198	1 ⊢ (∀𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1533  ∃wex 1774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-10 2137  ax-12 2173

This theorem depends on definitions:  df-bi 207  df-ex 1775  df-nf 1779

This theorem is referenced by: (None)