Theorem bj-modal4 37400

Description: First-order logic form of the modal axiom (4). See hba1 2330. This is the standard proof of the implication in modal logic (B5 ⇒ 4). Its dual statement is bj-modal4e 37401. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-modal4	⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Proof of Theorem bj-modal4

Step	Hyp	Ref	Expression
1		bj-modalbe 37372	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥∃𝑥∀𝑥𝜑)
2		hbe1a 2182	. 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
3	1, 2	sylg 1856	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bj-modal4e  37401  bj-substax12  37408  bj-nnfa1  37468