Theorem bj-exalbial 34038

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 2155	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	19.9 2205	1 ⊢ (∃𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 208  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)