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Theorem equsb3ALT 2431
Description: Alternate proof of equsb3 2430, shorter but requiring ax-11 2032. (Contributed by Raph Levien and FL, 4-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
equsb3ALT ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Distinct variable group:   𝑦,𝑧

Proof of Theorem equsb3ALT
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equsb3lem 2429 . . 3 ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧)
21sbbii 1885 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧)
3 nfv 1841 . . 3 𝑤 𝑦 = 𝑧
43sbco2 2413 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
5 equsb3lem 2429 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
62, 4, 53bitr3i 290 1 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879
This theorem is referenced by: (None)
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