Theorem ralxfr2d 5357

Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.)

Hypotheses

Ref	Expression
ralxfr2d.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉)
ralxfr2d.2	⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴))
ralxfr2d.3	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ralxfr2d	⊢ (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)   𝑉(𝑥,𝑦)

Proof of Theorem ralxfr2d

Step	Hyp	Ref	Expression
1		ralxfr2d.1	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉)
2		elisset 2819	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 𝑥 = 𝐴)
4		ralxfr2d.2	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴))
5	4	biimprd 248	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
6		r19.23v 3165	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐶 (𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
7	5, 6	sylibr 234	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
8	7	r19.21bi 3230	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
9		eleq1 2825	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
10	8, 9	mpbidi 241	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
11	10	exlimdv 1935	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
12	3, 11	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
13	4	biimpa 476	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
14		ralxfr2d.3	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
15	12, 13, 14	ralxfrd 5355	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063

This theorem is referenced by:  rexxfr2d  5358  ralrn  7042  ralima  7193  ralimaOLD  7196  cnrest2  23242  cnprest2  23246  connsuba  23376  subislly  23437  trfbas2  23799  trfil2  23843  flimrest  23939  fclsrest  23980  tsmssubm  24099  metucn  24527  ist0cld  34011  extoimad  44520