Theorem equs5a 1722

Description: A property related to substitution that unlike equs5 1757 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Assertion

Ref	Expression
equs5a	⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Proof of Theorem equs5a

Step	Hyp	Ref	Expression
1		hba1 1478	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑦 → 𝜑))
2		ax-11 1442	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	imp 122	. 2 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	1, 3	exlimih 1529	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1287  ∃wex 1426

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-gen 1383  ax-ie2 1428  ax-11 1442  ax-ial 1472

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  equs5e  1723  sb4a  1729  equs45f  1730