Theorem equs5a 1794

Description: A property related to substitution that unlike equs5 1829 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 1540	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑦 → 𝜑))
2		ax-11 1506	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	imp 124	. 2 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	1, 3	exlimih 1593	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1351  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-gen 1449  ax-ie2 1494  ax-11 1506  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  equs5e  1795  sb4a  1801  equs45f  1802