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Theorem equsb3ALT 2421
 Description: Alternate proof of equsb3 2420, shorter but requiring ax-11 2021. (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 2419 . . 3 ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧)
21sbbii 1874 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧)
3 nfv 1830 . . 3 𝑤 𝑦 = 𝑧
43sbco2 2403 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
5 equsb3lem 2419 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
62, 4, 53bitr3i 289 1 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by: (None)
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