Theorem bj-alexbiex 34783

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 2153	. 2 ⊢ Ⅎ𝑥∃𝑥𝜑
2	1	19.3 2202	1 ⊢ (∀𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 209  ∀wal 1541  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2143  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-ex 1788  df-nf 1792

This theorem is referenced by: (None)