Theorem ralxfrALT 5415

Description: Alternate proof of ralxfr 5414 which does not use ralxfrd 5408. (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 3618	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))
4	1, 3	syl 17	. . . 4 ⊢ (𝑦 ∈ 𝐶 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))
5	4	com12 32	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐶 → 𝜓))
6	5	ralrimiv 3145	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐶 𝜓)
7		ralxfr.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
8		nfra1 3284	. . . . 5 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐶 𝜓
9		nfv 1914	. . . . 5 ⊢ Ⅎ𝑦𝜑
10		rsp 3247	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (𝑦 ∈ 𝐶 → 𝜓))
11	2	biimprcd 250	. . . . . 6 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑))
12	10, 11	syl6 35	. . . . 5 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (𝑦 ∈ 𝐶 → (𝑥 = 𝐴 → 𝜑)))
13	8, 9, 12	rexlimd 3266	. . . 4 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 → 𝜑))
14	7, 13	syl5 34	. . 3 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (𝑥 ∈ 𝐵 → 𝜑))
15	14	ralrimiv 3145	. 2 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → ∀𝑥 ∈ 𝐵 𝜑)
16	6, 15	impbii 209	1 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071

This theorem is referenced by: (None)