Theorem ralxfrALT 4887

Description: Alternate proof of ralxfr 4886 which does not use ralxfrd 4879. (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 3305	. . . . 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 2965	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐶 𝜓)
7		ralxfr.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
8		nfra1 2941	. . . . 5 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐶 𝜓
9		nfv 1843	. . . . 5 ⊢ Ⅎ𝑦𝜑
10		rsp 2929	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (𝑦 ∈ 𝐶 → 𝜓))
11	2	biimprcd 240	. . . . . 6 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑))
12	10, 11	syl6 35	. . . . 5 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (𝑦 ∈ 𝐶 → (𝑥 = 𝐴 → 𝜑)))
13	8, 9, 12	rexlimd 3026	. . . 4 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 → 𝜑))
14	7, 13	syl5 34	. . 3 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → (𝑥 ∈ 𝐵 → 𝜑))
15	14	ralrimiv 2965	. 2 ⊢ (∀𝑦 ∈ 𝐶 𝜓 → ∀𝑥 ∈ 𝐵 𝜑)
16	6, 15	impbii 199	1 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by: (None)