Theorem nfmodv 2587

Description: Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2589 for a version without disjoint variable conditions but requiring ax-13 2404. (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 2568	. 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
2		nfv 1944	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfmodv.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfmodv.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5		nfvd 1945	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
6	4, 5	nfimd 1924	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝑦 = 𝑧))
7	3, 6	nfald 2361	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 → 𝑦 = 𝑧))
8	2, 7	nfexd 2362	. 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
9	1, 8	nfxfrd 1884	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1568  ∃wex 1809  Ⅎwnf 1813  ∃*wmo 2565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-nf 1814  df-mo 2567

This theorem is referenced by:  nfmov  2588  nfeudw  2619  nfdisjw  5089  wl-mo3t  38212