Theorem nfeuw 2591

Description: Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2592 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by NM, 8-Mar-1995.) Avoid ax-13 2375. (Revised by GG, 10-Jan-2024.)

Hypothesis

Ref	Expression
nfeuw.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfeuw	⊢ Ⅎ𝑥∃!𝑦𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem nfeuw

Step	Hyp	Ref	Expression
1		nftru 1801	. . 3 ⊢ Ⅎ𝑦⊤
2		nfeuw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfeudw 2589	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦𝜑)
5	4	mptru 1544	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

Syntax hints:  ⊤wtru 1538  Ⅎwnf 1780  ∃!weu 2566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-mo 2538  df-eu 2567

This theorem is referenced by:  nfreuw  3412  eusv2nf  5401  reusv2lem3  5406  bnj1489  35049  setrec2  48926