Theorem ralxfr2d 5372

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 2842	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 𝑥 = 𝐴)
4		ralxfr2d.2	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴))
5	4	biimprd 251	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
6		r19.23v 3189	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐶 (𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
7	5, 6	sylibr 237	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
8	7	r19.21bi 3254	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
9		eleq1 2848	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
10	8, 9	mpbidi 244	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
11	10	exlimdv 1965	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
12	3, 11	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
13	4	biimpa 482	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
14		ralxfr2d.3	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
15	12, 13, 14	ralxfrd 5370	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087

This theorem is used by:  rexxfr2d  5373  ralrn  7093  ralima  7247  cnrest2  23613  cnprest2  23617  connsuba  23747  subislly  23809  trfbas2  24171  trfil2  24215  flimrest  24311  fclsrest  24352  tsmssubm  24471  metucn  24899  ist0cld  34515  extoimad  45301