Theorem eu3h 2064

Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)

Hypothesis

Ref	Expression
eu3h.1	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

Ref	Expression
eu3h	⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eu3h

Step	Hyp	Ref	Expression
1		euex 2049	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
2		eu3h.1	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	eumo0 2050	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
4	1, 3	jca 304	. 2 ⊢ (∃!𝑥𝜑 → (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
5	2	nfi 1455	. . . . 5 ⊢ Ⅎ𝑦𝜑
6	5	mo23 2060	. . . 4 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
7	6	anim2i 340	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8	5	eu2 2063	. . 3 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
9	7, 8	sylibr 133	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → ∃!𝑥𝜑)
10	4, 9	impbii 125	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346  ∃wex 1485  [wsb 1755  ∃!weu 2019

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022

This theorem is referenced by:  eu3  2065  mo2r  2071  2eu4  2112