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Theorem sbalv 2492
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 2490 . 2 ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧[𝑦 / 𝑥]𝜑)
2 sbalv.1 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
32albii 1787 . 2 (∀𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝜓)
41, 3bitri 264 1 ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1521  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  sbmo  2544  sbabel  2822  mo5f  29452
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