Theorem hbaes 1648

Description: Rule that applies hbae 1646 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 1646	. 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 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)