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Theorem equsb3 1841
Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
Assertion
Ref Expression
equsb3 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Distinct variable group:   𝑦,𝑧

Proof of Theorem equsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equsb3lem 1840 . . 3 ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧)
21sbbii 1664 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧)
3 ax-17 1435 . . 3 (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
43sbco2v 1837 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
5 equsb3lem 1840 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
62, 4, 53bitr3i 203 1 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Colors of variables: wff set class
Syntax hints:  wb 102  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  sb8eu  1929  sb8euh  1939  sb8iota  4902
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