Theorem reuxfr1 3706

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5356 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 481	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
3		reuxfr1.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
4	3	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
5		reuxfr1.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	2, 4, 5	reuxfr1ds 3705	. 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓))
7	6	mptru 1548	1 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541  ⊤wtru 1542   ∈ wcel 2111  ∃!wreu 3344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347

This theorem is referenced by:  zmax  12843  rebtwnz  12845