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Theorem equsb3 1924
 Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
Assertion
Ref Expression
equsb3 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
Distinct variable group:   𝑥,𝑧

Proof of Theorem equsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equsb3lem 1923 . . 3 ([𝑤 / 𝑥]𝑥 = 𝑧𝑤 = 𝑧)
21sbbii 1738 . 2 ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝑧 ↔ [𝑦 / 𝑤]𝑤 = 𝑧)
3 ax-17 1506 . . 3 (𝑥 = 𝑧 → ∀𝑤 𝑥 = 𝑧)
43sbco2vh 1918 . 2 ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝑧 ↔ [𝑦 / 𝑥]𝑥 = 𝑧)
5 equsb3lem 1923 . 2 ([𝑦 / 𝑤]𝑤 = 𝑧𝑦 = 𝑧)
62, 4, 53bitr3i 209 1 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  [wsb 1735 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736 This theorem is referenced by:  sb8eu  2012  sb8euh  2022  sb8iota  5095
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