Theorem subsym1 31596

Description: A symmetry with [𝑥 / 𝑦].

See negsym1 31586 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

Step	Hyp	Ref	Expression
1		fal 1482	. . . . . . . . . 10 ⊢ ¬ ⊥
2	1	intnan 951	. . . . . . . . 9 ⊢ ¬ (𝑦 = 𝑥 ∧ ⊥)
3	2	nex 1722	. . . . . . . 8 ⊢ ¬ ∃𝑦(𝑦 = 𝑥 ∧ ⊥)
4	3	intnan 951	. . . . . . 7 ⊢ ¬ ((𝑦 = 𝑥 → ⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ ⊥))
5		df-sb 1868	. . . . . . 7 ⊢ ([𝑥 / 𝑦]⊥ ↔ ((𝑦 = 𝑥 → ⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ ⊥)))
6	4, 5	mtbir 312	. . . . . 6 ⊢ ¬ [𝑥 / 𝑦]⊥
7	6	intnan 951	. . . . 5 ⊢ ¬ (𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥)
8	7	nex 1722	. . . 4 ⊢ ¬ ∃𝑦(𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥)
9	8	intnan 951	. . 3 ⊢ ¬ ((𝑦 = 𝑥 → [𝑥 / 𝑦]⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥))
10		df-sb 1868	. . 3 ⊢ ([𝑥 / 𝑦][𝑥 / 𝑦]⊥ ↔ ((𝑦 = 𝑥 → [𝑥 / 𝑦]⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥)))
11	9, 10	mtbir 312	. 2 ⊢ ¬ [𝑥 / 𝑦][𝑥 / 𝑦]⊥
12	11	pm2.21i 115	1 ⊢ ([𝑥 / 𝑦][𝑥 / 𝑦]⊥ → [𝑥 / 𝑦]𝜑)

Syntax hints:   → wi 4   ∧ wa 383  ⊥wfal 1480  ∃wex 1695  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713

This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-sb 1868

This theorem is referenced by: (None)