Theorem cbveud 34184

Description: Deduction used to change bound variables in an existential uniqueness quantifier, using implicit substitution. (Contributed by ML, 27-Mar-2021.)

Hypotheses

Ref	Expression
cbveud.1	⊢ Ⅎ𝑥𝜑
cbveud.2	⊢ Ⅎ𝑦𝜑
cbveud.3	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbveud.4	⊢ (𝜑 → Ⅎ𝑥𝜒)
cbveud.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbveud	⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem cbveud

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbveud.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		cbveud.2	. . . 4 ⊢ Ⅎ𝑦𝜑
3		cbveud.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		nfvd 1893	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦 𝑥 = 𝑧)
5	3, 4	nfbid 1884	. . . 4 ⊢ (𝜑 → Ⅎ𝑦(𝜓 ↔ 𝑥 = 𝑧))
6		cbveud.4	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
7		nfvd 1893	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
8	6, 7	nfbid 1884	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝜒 ↔ 𝑦 = 𝑧))
9		cbveud.5	. . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
10		simpr 485	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (𝜓 ↔ 𝜒)) → (𝜓 ↔ 𝜒))
11		equequ1 2009	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
12	11	adantr 481	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (𝜓 ↔ 𝜒)) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
13	10, 12	bibi12d 347	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ (𝜓 ↔ 𝜒)) → ((𝜓 ↔ 𝑥 = 𝑧) ↔ (𝜒 ↔ 𝑦 = 𝑧)))
14	13	ex 413	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜓 ↔ 𝜒) → ((𝜓 ↔ 𝑥 = 𝑧) ↔ (𝜒 ↔ 𝑦 = 𝑧))))
15	9, 14	sylcom 30	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → ((𝜓 ↔ 𝑥 = 𝑧) ↔ (𝜒 ↔ 𝑦 = 𝑧))))
16	1, 2, 5, 8, 15	cbv2 2379	. . 3 ⊢ (𝜑 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ↔ ∀𝑦(𝜒 ↔ 𝑦 = 𝑧)))
17	16	exbidv 1899	. 2 ⊢ (𝜑 → (∃𝑧∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜒 ↔ 𝑦 = 𝑧)))
18		eu6 2617	. 2 ⊢ (∃!𝑥𝜓 ↔ ∃𝑧∀𝑥(𝜓 ↔ 𝑥 = 𝑧))
19		eu6 2617	. 2 ⊢ (∃!𝑦𝜒 ↔ ∃𝑧∀𝑦(𝜒 ↔ 𝑦 = 𝑧))
20	17, 18, 19	3bitr4g 315	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520   = wceq 1522  ∃wex 1761  Ⅎwnf 1765  ∃!weu 2611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-mo 2576  df-eu 2612

This theorem is referenced by:  cbvreud  34185