Theorem rexxfrd 4384

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 1508	. . . . 5 ⊢ Ⅎ𝑦𝜓
2	1	19.3 1533	. . . 4 ⊢ (∀𝑦𝜓 ↔ 𝜓)
3		ralxfrd.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
4		df-rex 2422	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))
5		19.29 1599	. . . . . . . . . 10 ⊢ ((∀𝑦𝜓 ∧ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) → ∃𝑦(𝜓 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)))
6		an12 550	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) ↔ (𝑦 ∈ 𝐶 ∧ (𝜓 ∧ 𝑥 = 𝐴)))
7	6	exbii 1584	. . . . . . . . . 10 ⊢ (∃𝑦(𝜓 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝜓 ∧ 𝑥 = 𝐴)))
8	5, 7	sylib 121	. . . . . . . . 9 ⊢ ((∀𝑦𝜓 ∧ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) → ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝜓 ∧ 𝑥 = 𝐴)))
9		df-rex 2422	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝜓 ∧ 𝑥 = 𝐴)))
10	8, 9	sylibr 133	. . . . . . . 8 ⊢ ((∀𝑦𝜓 ∧ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) → ∃𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴))
11	4, 10	sylan2b 285	. . . . . . 7 ⊢ ((∀𝑦𝜓 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → ∃𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴))
12		ralxfrd.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
13	12	biimpd 143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
14	13	expimpd 360	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝜓) → 𝜒))
15	14	ancomsd 267	. . . . . . . 8 ⊢ (𝜑 → ((𝜓 ∧ 𝑥 = 𝐴) → 𝜒))
16	15	reximdv 2533	. . . . . . 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 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑦𝜓 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → ∃𝑦 ∈ 𝐶 𝜒))
19	3, 18	mpan2d 424	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑦𝜓 → ∃𝑦 ∈ 𝐶 𝜒))
20	2, 19	syl5bir 152	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 → ∃𝑦 ∈ 𝐶 𝜒))
21	20	rexlimdva 2549	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → ∃𝑦 ∈ 𝐶 𝜒))
22		ralxfrd.1	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
23	12	adantlr 468	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
24	22, 23	rspcedv 2793	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))
25	24	rexlimdva 2549	. 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 𝜒 → ∃𝑥 ∈ 𝐵 𝜓))
26	21, 25	impbid 128	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688

This theorem is referenced by:  rexxfr2d  4386  rexxfr  4389