Theorem sb8mo 2601

Description: Variable substitution for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sb8eu.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb8mo	⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8mo

Step	Hyp	Ref	Expression
1		sb8eu.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	sb8e 2522	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3	1	sb8eu 2600	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
4	2, 3	imbi12i 350	. 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
5		moeu 2583	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
6		moeu 2583	. 2 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
7	4, 5, 6	3bitr4i 302	1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 205  ∃wex 1783  Ⅎwnf 1787  [wsb 2068  ∃*wmo 2538  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569

This theorem is referenced by:  cbvmo  2605