Theorem nfmodv 2559

Description: Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2561 for a version without disjoint variable conditions but requiring ax-13 2372. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by BJ, 28-Jan-2023.)

Hypotheses

Ref	Expression
nfmodv.1	⊢ Ⅎ𝑦𝜑
nfmodv.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfmodv	⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfmodv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mo 2540	. 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
2		nfv 1917	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfmodv.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfmodv.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5		nfvd 1918	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
6	4, 5	nfimd 1897	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝑦 = 𝑧))
7	3, 6	nfald 2322	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 → 𝑦 = 𝑧))
8	2, 7	nfexd 2323	. 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
9	1, 8	nfxfrd 1856	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786  ∃*wmo 2538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787  df-mo 2540

This theorem is referenced by:  nfmov  2560  nfeudw  2591  nfrmowOLD  3307  nfdisjw  5051