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Theorem sbalyz 2011
Description: Move universal quantifier in and out of substitution. Identical to sbal 2012 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 1552 . . . 4 𝑥𝑥𝜑
21nfsbxy 1954 . . 3 𝑥[𝑧 / 𝑦]∀𝑥𝜑
3 ax-4 1521 . . . 4 (∀𝑥𝜑𝜑)
43sbimi 1775 . . 3 ([𝑧 / 𝑦]∀𝑥𝜑 → [𝑧 / 𝑦]𝜑)
52, 4alrimi 1533 . 2 ([𝑧 / 𝑦]∀𝑥𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
6 sb6 1898 . . . . 5 ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧𝜑))
76albii 1481 . . . 4 (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥𝑦(𝑦 = 𝑧𝜑))
8 alcom 1489 . . . 4 (∀𝑥𝑦(𝑦 = 𝑧𝜑) ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑))
97, 8bitri 184 . . 3 (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑))
10 nfv 1539 . . . . . 6 𝑥 𝑦 = 𝑧
1110stdpc5 1595 . . . . 5 (∀𝑥(𝑦 = 𝑧𝜑) → (𝑦 = 𝑧 → ∀𝑥𝜑))
1211alimi 1466 . . . 4 (∀𝑦𝑥(𝑦 = 𝑧𝜑) → ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑))
13 sb2 1778 . . . 4 (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) → [𝑧 / 𝑦]∀𝑥𝜑)
1412, 13syl 14 . . 3 (∀𝑦𝑥(𝑦 = 𝑧𝜑) → [𝑧 / 𝑦]∀𝑥𝜑)
159, 14sylbi 121 . 2 (∀𝑥[𝑧 / 𝑦]𝜑 → [𝑧 / 𝑦]∀𝑥𝜑)
165, 15impbii 126 1 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1362  [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  sbal  2012
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