Theorem sb8eulem 2598

Description: Lemma. Factor out the common proof skeleton of sb8euv 2599 and sb8eu 2600. Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) Factor out common proof lines. (Revised by Wolf Lammen, 9-Feb-2023.)

Hypothesis

Ref	Expression
sb8eulem.nfsb	⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑

Assertion

Ref	Expression
sb8eulem	⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Distinct variable groups:   𝑦,𝑤   𝜑,𝑤   𝑥,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb8eulem

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1918	. . . . 5 ⊢ Ⅎ𝑤(𝜑 ↔ 𝑥 = 𝑧)
2	1	sb8v 2352	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
3		equsb3 2103	. . . . . 6 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
4	3	sblbis 2309	. . . . 5 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧))
5	4	albii 1823	. . . 4 ⊢ (∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧))
6		sb8eulem.nfsb	. . . . . 6 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
7		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑦 𝑤 = 𝑧
8	6, 7	nfbi 1907	. . . . 5 ⊢ Ⅎ𝑦([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧)
9		nfv 1918	. . . . 5 ⊢ Ⅎ𝑤([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)
10		sbequ 2087	. . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
11		equequ1 2029	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑦 = 𝑧))
12	10, 11	bibi12d 345	. . . . 5 ⊢ (𝑤 = 𝑦 → (([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)))
13	8, 9, 12	cbvalv1 2340	. . . 4 ⊢ (∀𝑤([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
14	2, 5, 13	3bitri 296	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
15	14	exbii 1851	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
16		eu6 2574	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
17		eu6 2574	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
18	15, 16, 17	3bitr4i 302	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1537  ∃wex 1783  Ⅎwnf 1787  [wsb 2068  ∃!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

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:  sb8euv  2599  sb8eu  2600