Theorem 2eu2ex 2030

Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Assertion

Ref	Expression
2eu2ex	⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)

Proof of Theorem 2eu2ex

Step	Hyp	Ref	Expression
1		euex 1971	. 2 ⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃!𝑦𝜑)
2		euex 1971	. . 3 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
3	2	eximi 1531	. 2 ⊢ (∃𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)
4	1, 3	syl 14	1 ⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)

Syntax hints:   → wi 4  ∃wex 1421  ∃!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944

This theorem is referenced by: (None)