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Theorem sbalv 1981
Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
sbalv.1 ([𝑦 / 𝑥]𝜑𝜓)
Assertion
Ref Expression
sbalv ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem sbalv
StepHypRef Expression
1 sbal 1976 . 2 ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧[𝑦 / 𝑥]𝜑)
2 sbalv.1 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
32albii 1447 . 2 (∀𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝜓)
41, 3bitri 183 1 ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1330  [wsb 1736
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737
This theorem is referenced by:  sbmo  2059  sbabel  2308  peano2  4516
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