Theorem rexxfrd 4551

Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)

Hypotheses

Ref	Expression
ralxfrd.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
ralxfrd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
ralxfrd.3	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexxfrd	⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))

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

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)   𝐶(𝑦)

Proof of Theorem rexxfrd

Step	Hyp	Ref	Expression
1		nfv 1574	. . . . 5 ⊢ Ⅎ𝑦𝜓
2	1	19.3 1600	. . . 4 ⊢ (∀𝑦𝜓 ↔ 𝜓)
3		ralxfrd.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
4		df-rex 2514	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))
5		19.29 1666	. . . . . . . . . 10 ⊢ ((∀𝑦𝜓 ∧ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) → ∃𝑦(𝜓 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)))
6		an12 561	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) ↔ (𝑦 ∈ 𝐶 ∧ (𝜓 ∧ 𝑥 = 𝐴)))
7	6	exbii 1651	. . . . . . . . . 10 ⊢ (∃𝑦(𝜓 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝜓 ∧ 𝑥 = 𝐴)))
8	5, 7	sylib 122	. . . . . . . . 9 ⊢ ((∀𝑦𝜓 ∧ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) → ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝜓 ∧ 𝑥 = 𝐴)))
9		df-rex 2514	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝜓 ∧ 𝑥 = 𝐴)))
10	8, 9	sylibr 134	. . . . . . . 8 ⊢ ((∀𝑦𝜓 ∧ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) → ∃𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴))
11	4, 10	sylan2b 287	. . . . . . 7 ⊢ ((∀𝑦𝜓 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → ∃𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴))
12		ralxfrd.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
13	12	biimpd 144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
14	13	expimpd 363	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝜓) → 𝜒))
15	14	ancomsd 269	. . . . . . . 8 ⊢ (𝜑 → ((𝜓 ∧ 𝑥 = 𝐴) → 𝜒))
16	15	reximdv 2631	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴) → ∃𝑦 ∈ 𝐶 𝜒))
17	11, 16	syl5 32	. . . . . 6 ⊢ (𝜑 → ((∀𝑦𝜓 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → ∃𝑦 ∈ 𝐶 𝜒))
18	17	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑦𝜓 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → ∃𝑦 ∈ 𝐶 𝜒))
19	3, 18	mpan2d 428	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑦𝜓 → ∃𝑦 ∈ 𝐶 𝜒))
20	2, 19	biimtrrid 153	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 → ∃𝑦 ∈ 𝐶 𝜒))
21	20	rexlimdva 2648	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → ∃𝑦 ∈ 𝐶 𝜒))
22		ralxfrd.1	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
23	12	adantlr 477	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
24	22, 23	rspcedv 2911	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))
25	24	rexlimdva 2648	. 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 𝜒 → ∃𝑥 ∈ 𝐵 𝜓))
26	21, 25	impbid 129	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1393   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801

This theorem is referenced by:  rexxfr2d  4553  rexxfr  4556