Theorem sbnf 2312

Description: Move nonfree predicate in and out of substitution; see sbal 2169 and sbex 2281. (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 1784	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2	1	sbbii 2076	. 2 ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ [𝑧 / 𝑦](∃𝑥𝜑 → ∀𝑥𝜑))
3		sbim 2303	. 2 ⊢ ([𝑧 / 𝑦](∃𝑥𝜑 → ∀𝑥𝜑) ↔ ([𝑧 / 𝑦]∃𝑥𝜑 → [𝑧 / 𝑦]∀𝑥𝜑))
4		sbex 2281	. . . 4 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
5		sbal 2169	. . . 4 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
6	4, 5	imbi12i 350	. . 3 ⊢ (([𝑧 / 𝑦]∃𝑥𝜑 → [𝑧 / 𝑦]∀𝑥𝜑) ↔ (∃𝑥[𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑))
7		df-nf 1784	. . 3 ⊢ (Ⅎ𝑥[𝑧 / 𝑦]𝜑 ↔ (∃𝑥[𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑))
8	6, 7	bitr4i 278	. 2 ⊢ (([𝑧 / 𝑦]∃𝑥𝜑 → [𝑧 / 𝑦]∀𝑥𝜑) ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)
9	2, 3, 8	3bitri 297	1 ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065

This theorem is referenced by:  bj-nfcf  36941  wl-nfsbtv  37595