Theorem bj-biexal3 36720

Description: When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)

Assertion

Ref	Expression
bj-biexal3	⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑 → 𝜓))

Proof of Theorem bj-biexal3

Step	Hyp	Ref	Expression
1		bj-biexal1 36718	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
2		bj-biexal2 36719	. 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
3	1, 2	bitr4i 278	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539  ∃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 1968  ax-7 2009  ax-10 2143  ax-12 2179

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)