Theorem sb8euv 2603

Description: Variable substitution in unique existential quantifier. Version of sb8eu 2604 requiring more disjoint variables, but fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Wolf Lammen, 7-Feb-2023.)

Hypothesis

Ref	Expression
sb8euv.nf	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb8euv	⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb8euv

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sb8euv.nf	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	nfsbv 2339	. 2 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
3	2	sb8eulem 2602	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 207  Ⅎwnf 1790  [wsb 2073  ∃!weu 2572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573

This theorem is referenced by:  eu1  2614