Theorem bj-hbaeb 37482

Description: Biconditional version of hbae 2462. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-hbaeb	⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)

Proof of Theorem bj-hbaeb

Step	Hyp	Ref	Expression
1		bj-hbaeb2 37481	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦)
2		alcom 2193	. 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)
3	1, 2	bitri 278	1 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813

This theorem is used by: (None)