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Theorem subsym1 33889
 Description: A symmetry with [𝑥 / 𝑦]. See negsym1 33879 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 fal 1552 . . . . . . . . . 10 ¬ ⊥
21intnan 490 . . . . . . . . 9 ¬ (𝑥 = 𝑦 ∧ ⊥)
32nex 1802 . . . . . . . 8 ¬ ∃𝑥(𝑥 = 𝑦 ∧ ⊥)
43intnan 490 . . . . . . 7 ¬ ((𝑥 = 𝑦 → ⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ⊥))
5 dfsb1 2502 . . . . . . 7 ([𝑦 / 𝑥]⊥ ↔ ((𝑥 = 𝑦 → ⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ⊥)))
64, 5mtbir 326 . . . . . 6 ¬ [𝑦 / 𝑥]⊥
76intnan 490 . . . . 5 ¬ (𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥)
87nex 1802 . . . 4 ¬ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥)
98intnan 490 . . 3 ¬ ((𝑥 = 𝑦 → [𝑦 / 𝑥]⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥))
10 dfsb1 2502 . . 3 ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ ↔ ((𝑥 = 𝑦 → [𝑦 / 𝑥]⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥)))
119, 10mtbir 326 . 2 ¬ [𝑦 / 𝑥][𝑦 / 𝑥]⊥
1211pm2.21i 119 1 ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ⊥wfal 1550  ∃wex 1781  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176  ax-13 2382 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by: (None)
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