Theorem nfmodv 2589

Description: Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2591 for a version without disjoint variable conditions but requiring ax-13 2406. (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		dfmo 2570	. 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
2		nfv 1947	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfmodv.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfmodv.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5		nfvd 1948	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
6	4, 5	nfimd 1927	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝑦 = 𝑧))
7	3, 6	nfald 2363	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 → 𝑦 = 𝑧))
8	2, 7	nfexd 2364	. 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
9	1, 8	nfxfrd 1887	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816  ∃*wmo 2567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-mo 2569

This theorem is used by:  nfmov  2590  nfeudw  2621  nfdisjw  5090  wl-mo3t  38264