Theorem bj-hbaeb 36993

Description: Biconditional version of hbae 2434. (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 36992	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦)
2		alcom 2165	. 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)
3	1, 2	bitri 275	1 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)

Syntax hints:   ↔ wb 206  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-11 2163  ax-12 2183  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)