Theorem rexxfr 4166

Description: Transfer existence from a variable x to another variable y contained in expression A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)

Hypotheses

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

Assertion

Ref	Expression
rexxfr	⊢ (∃x ∈ B φ ↔ ∃y ∈ 𝐶 ψ)

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

Allowed substitution hints:   φ(x)   ψ(y)   A(y)   𝐶(y)

Proof of Theorem rexxfr

Step	Hyp	Ref	Expression
1		ralxfr.1	. . . 4 ⊢ (y ∈ 𝐶 → A ∈ B)
2	1	adantl 262	. . 3 ⊢ (( ⊤ ∧ y ∈ 𝐶) → A ∈ B)
3		ralxfr.2	. . . 4 ⊢ (x ∈ B → ∃y ∈ 𝐶 x = A)
4	3	adantl 262	. . 3 ⊢ (( ⊤ ∧ x ∈ B) → ∃y ∈ 𝐶 x = A)
5		ralxfr.3	. . . 4 ⊢ (x = A → (φ ↔ ψ))
6	5	adantl 262	. . 3 ⊢ (( ⊤ ∧ x = A) → (φ ↔ ψ))
7	2, 4, 6	rexxfrd 4161	. 2 ⊢ ( ⊤ → (∃x ∈ B φ ↔ ∃y ∈ 𝐶 ψ))
8	7	trud 1251	1 ⊢ (∃x ∈ B φ ↔ ∃y ∈ 𝐶 ψ)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ⊤ wtru 1243   ∈ wcel 1390  ∃wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553

This theorem is referenced by: (None)