Theorem 2reu2reu2 32537

Description: Double restricted existential uniqueness implies two nested restricted existential uniqueness. (Contributed by AV, 5-Jul-2023.)

Assertion

Ref	Expression
2reu2reu2	⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reu2reu2

Step	Hyp	Ref	Expression
1		df-2reu 32533	. 2 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
2		2rexreu 3719	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
3	1, 2	sylbi 217	1 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ∧ wa 395  ∃wrex 3059  ∃!wreu 3347  ∃!w2reu 32532

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-10 2147  ax-11 2163  ax-12 2183

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-mo 2538  df-eu 2568  df-clel 2810  df-nfc 2884  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-2reu 32533

This theorem is referenced by: (None)