Theorem sbal 2450

Description: Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)

Assertion

Ref	Expression
sbal	⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝑦   𝑥,𝑧

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

Proof of Theorem sbal

Step	Hyp	Ref	Expression
1		nfae 2304	. . . 4 ⊢ Ⅎ𝑦∀𝑥 𝑥 = 𝑧
2		axc16gb 2121	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝜑 ↔ ∀𝑥𝜑))
3	1, 2	sbbid 2391	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
4		axc16gb 2121	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
5	3, 4	bitr3d 269	. 2 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
6		sbal1 2448	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
7	5, 6	pm2.61i 175	1 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)

Syntax hints:   ↔ wb 195  ∀wal 1473  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868

This theorem is referenced by:  sbex  2451  sbalv  2452  sbcal  3452  ax11-pm2  32011  bj-sbnf  32016  sbcalgOLD  37773