Theorem sb8eu 2661

Description: Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2379. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2660. (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 2542	. 2 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
3	2	sb8eulem 2659	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 209  Ⅎwnf 1785  [wsb 2069  ∃!weu 2628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629

This theorem is referenced by:  sb8mo  2662  cbveu  2668  cbvreu  3394