Theorem sbalOLD 2574

Description: Obsolete version of sbal 2165 as of 13-Aug-2023. Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝑥,𝑧

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

Proof of Theorem sbalOLD

Step	Hyp	Ref	Expression
1		nfae 2454	. . . 4 ⊢ Ⅎ𝑦∀𝑥 𝑥 = 𝑧
2		axc16gb 2262	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝜑 ↔ ∀𝑥𝜑))
3	1, 2	sbbid 2245	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
4		axc16gb 2262	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
5	3, 4	bitr3d 283	. 2 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
6		sbal1 2571	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
7	5, 6	pm2.61i 184	1 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)

Syntax hints:   ↔ wb 208  ∀wal 1534  [wsb 2068

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 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069

This theorem is referenced by: (None)