Theorem ralxfrALT 4439

Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. This proof does not use ralxfrd 4434. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
ralxfrALT	⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓)

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

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

Proof of Theorem ralxfrALT

Step	Hyp	Ref	Expression
1		ralxfr.1	. . . . 5 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)
2		ralxfr.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	rspcv 2821	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))
4	1, 3	syl 14	. . . 4 ⊢ (𝑦 ∈ 𝐶 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))
5	4	com12 30	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐶 → 𝜓))
6	5	ralrimiv 2536	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐶 𝜓)
7		ralxfr.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
8		nfra1 2495	. . . . 5 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐶 𝜓
9		nfv 1515	. . . . 5 ⊢ Ⅎ𝑦𝜑
10		rsp 2511	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (𝑦 ∈ 𝐶 → 𝜓))
11	2	biimprcd 159	. . . . . 6 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑))
12	10, 11	syl6 33	. . . . 5 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (𝑦 ∈ 𝐶 → (𝑥 = 𝐴 → 𝜑)))
13	8, 9, 12	rexlimd 2578	. . . 4 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 → 𝜑))
14	7, 13	syl5 32	. . 3 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (𝑥 ∈ 𝐵 → 𝜑))
15	14	ralrimiv 2536	. 2 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → ∀𝑥 ∈ 𝐵 𝜑)
16	6, 15	impbii 125	1 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1342   ∈ wcel 2135  ∀wral 2442  ∃wrex 2443

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723

This theorem is referenced by: (None)