Theorem sb8eu 2590

Description: Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2367. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2589. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sb8eu.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb8eu	⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8eu

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sb8eu.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	nfsb 2518	. 2 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
3	2	sb8eulem 2588	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  Ⅎwnf 1778  [wsb 2060  ∃!weu 2558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2367

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559

This theorem is referenced by:  sb8mo  2591  cbveu  2599  cbvreu  3421