Theorem euexALTOLD 2643

Description: Obsolete proof of euex 2596 as of 31-Dec-2022. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
euexALTOLD	⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)

Proof of Theorem euexALTOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1873	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	eu1 2641	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
3		exsimpl 1831	. 2 ⊢ (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) → ∃𝑥𝜑)
4	2, 3	sylbi 209	1 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 387  ∀wal 1505  ∃wex 1742  [wsb 2015  ∃!weu 2583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-11 2093  ax-12 2106

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584

This theorem is referenced by: (None)