Theorem bj-exalbial 34621

Description: Adding a second quantifier is a tranparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.)

Assertion

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

Proof of Theorem bj-exalbial

Step	Hyp	Ref	Expression
1		nfa1 2152	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	19.9 2203	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 2141  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-or 848  df-ex 1788  df-nf 1792

This theorem is referenced by: (None)