Theorem hbaes 1744

Description: Rule that applies hbae 1742 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 1742	. 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 1371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)