Theorem alxfr 5330

Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.)

Hypothesis

Ref	Expression
alxfr.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
alxfr	⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

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

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem alxfr

Step	Hyp	Ref	Expression
1		alxfr.1	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	spcgv 3535	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))
3	2	com12 32	. . . . 5 ⊢ (∀𝑥𝜑 → (𝐴 ∈ 𝐵 → 𝜓))
4	3	alimdv 1919	. . . 4 ⊢ (∀𝑥𝜑 → (∀𝑦 𝐴 ∈ 𝐵 → ∀𝑦𝜓))
5	4	com12 32	. . 3 ⊢ (∀𝑦 𝐴 ∈ 𝐵 → (∀𝑥𝜑 → ∀𝑦𝜓))
6	5	adantr 481	. 2 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 → ∀𝑦𝜓))
7		nfa1 2148	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦𝜓
8		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑦𝜑
9		sp 2176	. . . . . . 7 ⊢ (∀𝑦𝜓 → 𝜓)
10	9, 1	syl5ibrcom 246	. . . . . 6 ⊢ (∀𝑦𝜓 → (𝑥 = 𝐴 → 𝜑))
11	7, 8, 10	exlimd 2211	. . . . 5 ⊢ (∀𝑦𝜓 → (∃𝑦 𝑥 = 𝐴 → 𝜑))
12	11	alimdv 1919	. . . 4 ⊢ (∀𝑦𝜓 → (∀𝑥∃𝑦 𝑥 = 𝐴 → ∀𝑥𝜑))
13	12	com12 32	. . 3 ⊢ (∀𝑥∃𝑦 𝑥 = 𝐴 → (∀𝑦𝜓 → ∀𝑥𝜑))
14	13	adantl 482	. 2 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑦𝜓 → ∀𝑥𝜑))
15	6, 14	impbid 211	1 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434

This theorem is referenced by: (None)