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Theorem equsb3ALT 2525
 Description: Alternate proof of equsb3 2524, shorter but requiring ax-11 2198. (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 2523 . . 3 ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧)
21sbbii 2068 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧)
3 nfv 2009 . . 3 𝑤 𝑦 = 𝑧
43sbco2 2506 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
5 equsb3lem 2523 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
62, 4, 53bitr3i 292 1 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 197  [wsb 2061 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352 This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062 This theorem is referenced by: (None)
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