Theorem reuxfr 3710

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
reuxfr.1	⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)
reuxfr.2	⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)

Assertion

Ref	Expression
reuxfr	⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑)

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

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfr

Step	Hyp	Ref	Expression
1		reuxfr.1	. . . 4 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)
2	1	adantl 485	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
3		reuxfr.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)
4	3	adantl 485	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)
5	2, 4	reuxfrd 3709	. 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑))
6	5	mptru 1566	1 ⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ⊤wtru 1560   ∈ wcel 2141  ∃wrex 3085  ∃!wreu 3364  ∃*wrmo 3365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367

This theorem is referenced by: (None)