Theorem sbnfOLD 2317

Description: Obsolete version of sbnf 2316 as of 2-May-2025. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝑥,𝑧

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

Proof of Theorem sbnfOLD

Step	Hyp	Ref	Expression
1		sbim 2307	. . . 4 ⊢ ([𝑧 / 𝑦](𝜑 → ∀𝑥𝜑) ↔ ([𝑧 / 𝑦]𝜑 → [𝑧 / 𝑦]∀𝑥𝜑))
2		sbal 2170	. . . . 5 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
3	2	imbi2i 336	. . . 4 ⊢ (([𝑧 / 𝑦]𝜑 → [𝑧 / 𝑦]∀𝑥𝜑) ↔ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑))
4	1, 3	bitri 275	. . 3 ⊢ ([𝑧 / 𝑦](𝜑 → ∀𝑥𝜑) ↔ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑))
5	4	albii 1817	. 2 ⊢ (∀𝑥[𝑧 / 𝑦](𝜑 → ∀𝑥𝜑) ↔ ∀𝑥([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑))
6		nf5 2286	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
7	6	sbbii 2076	. . 3 ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥(𝜑 → ∀𝑥𝜑))
8		sbal 2170	. . 3 ⊢ ([𝑧 / 𝑦]∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥[𝑧 / 𝑦](𝜑 → ∀𝑥𝜑))
9	7, 8	bitri 275	. 2 ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦](𝜑 → ∀𝑥𝜑))
10		nf5 2286	. 2 ⊢ (Ⅎ𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑))
11	5, 9, 10	3bitr4i 303	1 ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  Ⅎwnf 1781  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by: (None)