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Theorem subsym1 32551
Description: A symmetry with [𝑥 / 𝑦].

See negsym1 32541 for more information. (Contributed by Anthony Hart, 11-Sep-2011.)

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

Proof of Theorem subsym1
StepHypRef Expression
1 fal 1530 . . . . . . . . . 10 ¬ ⊥
21intnan 980 . . . . . . . . 9 ¬ (𝑦 = 𝑥 ∧ ⊥)
32nex 1771 . . . . . . . 8 ¬ ∃𝑦(𝑦 = 𝑥 ∧ ⊥)
43intnan 980 . . . . . . 7 ¬ ((𝑦 = 𝑥 → ⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ ⊥))
5 df-sb 1938 . . . . . . 7 ([𝑥 / 𝑦]⊥ ↔ ((𝑦 = 𝑥 → ⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ ⊥)))
64, 5mtbir 312 . . . . . 6 ¬ [𝑥 / 𝑦]⊥
76intnan 980 . . . . 5 ¬ (𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥)
87nex 1771 . . . 4 ¬ ∃𝑦(𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥)
98intnan 980 . . 3 ¬ ((𝑦 = 𝑥 → [𝑥 / 𝑦]⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥))
10 df-sb 1938 . . 3 ([𝑥 / 𝑦][𝑥 / 𝑦]⊥ ↔ ((𝑦 = 𝑥 → [𝑥 / 𝑦]⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥)))
119, 10mtbir 312 . 2 ¬ [𝑥 / 𝑦][𝑥 / 𝑦]⊥
1211pm2.21i 116 1 ([𝑥 / 𝑦][𝑥 / 𝑦]⊥ → [𝑥 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wfal 1528  wex 1744  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-sb 1938
This theorem is referenced by: (None)
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