Theorem sbnf 2348

Description: Move nonfree predicate in and out of substitution; see sbal 2209 and sbex 2318. (Contributed by BJ, 2-May-2019.) (Proof shortened by Wolf Lammen, 2-May-2025.)

Assertion

Ref	Expression
sbnf	⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝑦   𝑥,𝑧

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

Proof of Theorem sbnf

Step	Hyp	Ref	Expression
1		df-nf 1817	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2	1	sbbii 2115	. 2 ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ [𝑧 / 𝑦](∃𝑥𝜑 → ∀𝑥𝜑))
3		sbim 2340	. 2 ⊢ ([𝑧 / 𝑦](∃𝑥𝜑 → ∀𝑥𝜑) ↔ ([𝑧 / 𝑦]∃𝑥𝜑 → [𝑧 / 𝑦]∀𝑥𝜑))
4		sbex 2318	. . . 4 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
5		sbal 2209	. . . 4 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
6	4, 5	imbi12i 353	. . 3 ⊢ (([𝑧 / 𝑦]∃𝑥𝜑 → [𝑧 / 𝑦]∀𝑥𝜑) ↔ (∃𝑥[𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑))
7		df-nf 1817	. . 3 ⊢ (Ⅎ𝑥[𝑧 / 𝑦]𝜑 ↔ (∃𝑥[𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑))
8	6, 7	bitr4i 281	. 2 ⊢ (([𝑧 / 𝑦]∃𝑥𝜑 → [𝑧 / 𝑦]∀𝑥𝜑) ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)
9	2, 3, 8	3bitri 300	1 ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816  [wsb 2101

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 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2102

This theorem is used by:  bj-nfcf  38012  wl-nfsbtv  38700