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Theorem sbbib 2361
Description: Reversal of substitution. (Contributed by AV, 6-Aug-2023.) (Proof shortened by Wolf Lammen, 4-Sep-2023.)
Hypotheses
Ref Expression
sbbib.y 𝑦𝜑
sbbib.x 𝑥𝜓
Assertion
Ref Expression
sbbib (∀𝑦([𝑦 / 𝑥]𝜑𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbbib
StepHypRef Expression
1 nfs1v 2157 . . 3 𝑥[𝑦 / 𝑥]𝜑
2 sbbib.x . . 3 𝑥𝜓
31, 2nfbi 1910 . 2 𝑥([𝑦 / 𝑥]𝜑𝜓)
4 sbbib.y . . 3 𝑦𝜑
5 nfs1v 2157 . . 3 𝑦[𝑥 / 𝑦]𝜓
64, 5nfbi 1910 . 2 𝑦(𝜑 ↔ [𝑥 / 𝑦]𝜓)
7 sbequ12r 2249 . . 3 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
8 sbequ12 2248 . . 3 (𝑦 = 𝑥 → (𝜓 ↔ [𝑥 / 𝑦]𝜓))
97, 8bibi12d 346 . 2 (𝑦 = 𝑥 → (([𝑦 / 𝑥]𝜑𝜓) ↔ (𝜑 ↔ [𝑥 / 𝑦]𝜓)))
103, 6, 9cbvalv1 2342 1 (∀𝑦([𝑦 / 𝑥]𝜑𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1540  wnf 1790  [wsb 2071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-10 2141  ax-11 2158  ax-12 2175
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2072
This theorem is referenced by:  sbbibvv  2362  dfich2  44889
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