Theorem nfmodv 2585

Description: Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2587 for a version without disjoint variable conditions but requiring ax-13 2402. (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 2566	. 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
2		nfv 1933	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfmodv.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfmodv.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5		nfvd 1934	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
6	4, 5	nfimd 1913	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝑦 = 𝑧))
7	3, 6	nfald 2359	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 → 𝑦 = 𝑧))
8	2, 7	nfexd 2360	. 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
9	1, 8	nfxfrd 1873	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1557  ∃wex 1798  Ⅎwnf 1802  ∃*wmo 2563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-11 2190  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1799  df-nf 1803  df-mo 2565

This theorem is referenced by:  nfmov  2586  nfeudw  2617  nfdisjw  5076  wl-mo3t  38039