Theorem rexxfr2d 5358

Description: Transfer existential quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

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
rexxfr2d	⊢ (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 rexxfr2d

Step	Hyp	Ref	Expression
1		ralxfr2d.1	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉)
2		ralxfr2d.2	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴))
3		ralxfr2d.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
4	3	notbid 318	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (¬ 𝜓 ↔ ¬ 𝜒))
5	1, 2, 4	ralxfr2d 5357	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐶 ¬ 𝜒))
6	5	notbid 318	. 2 ⊢ (𝜑 → (¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐶 ¬ 𝜒))
7		dfrex2 3065	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓)
8		dfrex2 3065	. 2 ⊢ (∃𝑦 ∈ 𝐶 𝜒 ↔ ¬ ∀𝑦 ∈ 𝐶 ¬ 𝜒)
9	6, 7, 8	3bitr4g 314	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ 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:  rexrn  7041  reximaOLD  7195  cnpresti  23244  cnprest  23245  1stcrest  23409  subislly  23437  txrest  23587  trfil2  23843  met1stc  24477  metucn  24527  xrlimcnp  26946  esumlub  34237  esumfsup  34247  rexxfr3d  35851  ptrest  37867  djhcvat42  41788