Theorem reuxfr1 3723

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5395 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.)

Hypotheses

Ref	Expression
reuxfr1.1	⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)
reuxfr1.2	⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
reuxfr1.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
reuxfr1	⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓)

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

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

Proof of Theorem reuxfr1

Step	Hyp	Ref	Expression
1		reuxfr1.1	. . . 4 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)
2	1	adantl 486	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
3		reuxfr1.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
4	3	adantl 486	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
5		reuxfr1.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	2, 4, 5	reuxfr1ds 3722	. 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓))
7	6	mptru 1575	1 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1568  ⊤wtru 1569   ∈ wcel 2150  ∃!wreu 3374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377

This theorem is referenced by:  zmax  12972  rebtwnz  12974