Theorem sb8eu 2598

Description: Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2375. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2597. (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 2526	. 2 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
3	2	sb8eulem 2596	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 206  Ⅎwnf 1780  [wsb 2062  ∃!weu 2566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567

This theorem is referenced by:  sb8mo  2599  cbveu  2605  cbvreu  3425