Theorem hbaes 1608

Description: Rule that applies hbae 1606 to antecedent. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
hbalequs.1	⊢ (∀𝑧∀𝑥 𝑥 = 𝑦 → 𝜑)

Assertion

Ref	Expression
hbaes	⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)

Proof of Theorem hbaes

Step	Hyp	Ref	Expression
1		hbae 1606	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
2		hbalequs.1	. 2 ⊢ (∀𝑧∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	syl 14	1 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1241

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)