Theorem bj-exalbial 37141

Description: Adding a second quantifier over the same variable is a transparent 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 2184	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	19.9 2239	1 ⊢ (∃𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 208  ∀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  ax-7 2027  ax-10 2174  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1799  df-nf 1803

This theorem is referenced by: (None)