Theorem sb8euv 2597

Description: Variable substitution in unique existential quantifier. Version of sb8eu 2598 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 2329	. 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

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:  eu1  2608  cbvreuwOLD  3413