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Theorem eqsb3 2854
Description: Substitution applied to an atomic wff (class version of equsb3 2557). (Contributed by Rodolfo Medina, 28-Apr-2010.)
Assertion
Ref Expression
eqsb3 ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eqsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqsb3lem 2853 . . 3 ([𝑤 / 𝑦]𝑦 = 𝐴𝑤 = 𝐴)
21sbbii 2041 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑤]𝑤 = 𝐴)
3 nfv 1980 . . 3 𝑤 𝑦 = 𝐴
43sbco2 2540 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑦]𝑦 = 𝐴)
5 eqsb3lem 2853 . 2 ([𝑥 / 𝑤]𝑤 = 𝐴𝑥 = 𝐴)
62, 4, 53bitr3i 290 1 ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1620  [wsb 2034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-cleq 2741
This theorem is referenced by:  pm13.183  3472  eqsbc3  3604
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