Theorem 2eu2ex 2643

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 2577	. 2 ⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃!𝑦𝜑)
2		euex 2577	. . 3 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
3	2	eximi 1835	. 2 ⊢ (∃𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)
4	1, 3	syl 17	1 ⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)

Syntax hints:   → wi 4  ∃wex 1779  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-eu 2569

This theorem is referenced by:  2eu1  2651  2eu1v  2652