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Theorem subsym1 35312
Description: A symmetry with [𝑥 / 𝑦].

See negsym1 35302 for more information. (Contributed by Anthony Hart, 11-Sep-2011.)

Assertion
Ref Expression
subsym1 ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑)

Proof of Theorem subsym1
StepHypRef Expression
1 sbv 2092 . . 3 ([𝑦 / 𝑥]⊥ ↔ ⊥)
2 falim 1559 . . 3 (⊥ → 𝜑)
31, 2sylbi 216 . 2 ([𝑦 / 𝑥]⊥ → 𝜑)
43sbimi 2078 1 ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1554  [wsb 2068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972
This theorem depends on definitions:  df-bi 206  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069
This theorem is referenced by: (None)
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