Theorem bj-hbaeb 34929

Description: Biconditional version of hbae 2431. (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 34928	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦)
2		alcom 2158	. 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)
3	1, 2	bitri 274	1 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)

Syntax hints:   ↔ wb 205  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)