Theorem equs5e 1768

Description: A property related to substitution that unlike equs5 1802 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)

Assertion

Ref	Expression
equs5e	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem equs5e

Step	Hyp	Ref	Expression
1		19.8a 1570	. . . . 5 ⊢ (𝜑 → ∃𝑦𝜑)
2		hbe1 1472	. . . . 5 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
3	1, 2	syl 14	. . . 4 ⊢ (𝜑 → ∀𝑦∃𝑦𝜑)
4	3	anim2i 340	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦∃𝑦𝜑))
5	4	eximi 1580	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦∃𝑦𝜑))
6		equs5a 1767	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦∃𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
7	5, 6	syl 14	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332  ∃wex 1469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-11 1485  ax-4 1488  ax-ial 1515

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ax11e  1769  sb4e  1778