Theorem ralxfr2d 5409

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 2816	. . . 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 247	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
6		r19.23v 3183	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐶 (𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
7	5, 6	sylibr 233	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
8	7	r19.21bi 3249	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
9		eleq1 2822	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
10	8, 9	mpbidi 240	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
11	10	exlimdv 1937	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
12	3, 11	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
13	4	biimpa 478	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
14		ralxfr2d.3	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
15	12, 13, 14	ralxfrd 5407	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072

This theorem is referenced by:  rexxfr2d  5410  ralrn  7090  ralima  7240  cnrest2  22790  cnprest2  22794  connsuba  22924  subislly  22985  trfbas2  23347  trfil2  23391  flimrest  23487  fclsrest  23528  tsmssubm  23647  metucn  24080  ist0cld  32813  extoimad  42916