Theorem 2eu2ex 2125

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 2066	. 2 ⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃!𝑦𝜑)
2		euex 2066	. . 3 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
3	2	eximi 1610	. 2 ⊢ (∃𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)
4	1, 3	syl 14	1 ⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)

Syntax hints:   → wi 4  ∃wex 1502  ∃!weu 2036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-eu 2039

This theorem is referenced by: (None)