Theorem sbal 1993

Description: Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-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

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbalyz 1992	. . . 4 ⊢ ([𝑤 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑤 / 𝑦]𝜑)
2	1	sbbii 1758	. . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑤]∀𝑥[𝑤 / 𝑦]𝜑)
3		sbalyz 1992	. . 3 ⊢ ([𝑧 / 𝑤]∀𝑥[𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑)
4	2, 3	bitri 183	. 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑)
5		ax-17 1519	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑤∀𝑥𝜑)
6	5	sbco2vh 1938	. 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑)
7		ax-17 1519	. . . 4 ⊢ (𝜑 → ∀𝑤𝜑)
8	7	sbco2vh 1938	. . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑)
9	8	albii 1463	. 2 ⊢ (∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
10	4, 6, 9	3bitr3i 209	1 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)

Syntax hints:   ↔ wb 104  ∀wal 1346  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  sbal1  1995  sbalv  1998  sbcal  3006  sbcalg  3007