Theorem reuxfrd 3708

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 16-Jan-2012.) Separate variables 𝐵 and 𝐶. (Revised by Thierry Arnoux, 8-Oct-2017.)

Hypotheses

Ref	Expression
reuxfrd.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
reuxfrd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)

Assertion

Ref	Expression
reuxfrd	⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓))

Distinct variable groups:   𝑥,𝑦,𝜑   𝜓,𝑥   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Allowed substitution hints:   𝜓(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfrd

Step	Hyp	Ref	Expression
1		reuxfrd.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)
2		rmoan 3699	. . . . . . 7 ⊢ (∃*𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃*𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴))
3	1, 2	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴))
4		ancom 460	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜓))
5	4	rmobii 3360	. . . . . 6 ⊢ (∃*𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓))
6	3, 5	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓))
7	6	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓))
8		2reuswap 3706	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓)))
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓)))
10		2reuswap2 3707	. . . 4 ⊢ (∀𝑦 ∈ 𝐶 ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)) → (∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓)))
11		moeq 3667	. . . . . . 7 ⊢ ∃*𝑥 𝑥 = 𝐴
12	11	moani 2554	. . . . . 6 ⊢ ∃*𝑥((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴)
13		ancom 460	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)))
14		an12 646	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
15	13, 14	bitri 275	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
16	15	mobii 2549	. . . . . 6 ⊢ (∃*𝑥((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
17	12, 16	mpbi 230	. . . . 5 ⊢ ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓))
18	17	a1i 11	. . . 4 ⊢ (𝑦 ∈ 𝐶 → ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
19	10, 18	mprg 3058	. . 3 ⊢ (∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓))
20	9, 19	impbid1 225	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓)))
21		reuxfrd.1	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
22		biidd 262	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜓))
23	22	ceqsrexv 3611	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ 𝜓))
24	21, 23	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ 𝜓))
25	24	reubidva 3366	. 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓))
26	20, 25	bitrd 279	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃*wmo 2538  ∀wral 3052  ∃wrex 3062  ∃!wreu 3350  ∃*wrmo 3351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353

This theorem is referenced by:  reuxfr  3709  reuxfr1d  3710  reuxfr1dd  49163