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Theorem sbalyz 1950
Description: Move universal quantifier in and out of substitution. Identical to sbal 1951 except that it has an additional distinct variable constraint on 𝑦 and 𝑧. (Contributed by Jim Kingdon, 29-Dec-2017.)
Assertion
Ref Expression
sbalyz ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbalyz
StepHypRef Expression
1 nfa1 1504 . . . 4 𝑥𝑥𝜑
21nfsbxy 1893 . . 3 𝑥[𝑧 / 𝑦]∀𝑥𝜑
3 ax-4 1470 . . . 4 (∀𝑥𝜑𝜑)
43sbimi 1720 . . 3 ([𝑧 / 𝑦]∀𝑥𝜑 → [𝑧 / 𝑦]𝜑)
52, 4alrimi 1485 . 2 ([𝑧 / 𝑦]∀𝑥𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
6 sb6 1840 . . . . 5 ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧𝜑))
76albii 1429 . . . 4 (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥𝑦(𝑦 = 𝑧𝜑))
8 alcom 1437 . . . 4 (∀𝑥𝑦(𝑦 = 𝑧𝜑) ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑))
97, 8bitri 183 . . 3 (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑))
10 nfv 1491 . . . . . 6 𝑥 𝑦 = 𝑧
1110stdpc5 1546 . . . . 5 (∀𝑥(𝑦 = 𝑧𝜑) → (𝑦 = 𝑧 → ∀𝑥𝜑))
1211alimi 1414 . . . 4 (∀𝑦𝑥(𝑦 = 𝑧𝜑) → ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑))
13 sb2 1723 . . . 4 (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) → [𝑧 / 𝑦]∀𝑥𝜑)
1412, 13syl 14 . . 3 (∀𝑦𝑥(𝑦 = 𝑧𝜑) → [𝑧 / 𝑦]∀𝑥𝜑)
159, 14sylbi 120 . 2 (∀𝑥[𝑧 / 𝑦]𝜑 → [𝑧 / 𝑦]∀𝑥𝜑)
165, 15impbii 125 1 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312  [wsb 1718
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719
This theorem is referenced by:  sbal  1951
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