Theorem bj-hbaeb 37398

Description: Biconditional version of hbae 2461. (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 37397	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦)
2		alcom 2192	. 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)
3	1, 2	bitri 278	1 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)

Syntax hints:   ↔ wb 209  ∀wal 1566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812

This theorem is referenced by: (None)