Theorem subsym1 34616

Description: A symmetry with [𝑥 / 𝑦].

See negsym1 34606 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

Step	Hyp	Ref	Expression
1		fal 1553	. . . . . . . . . 10 ⊢ ¬ ⊥
2	1	intnan 487	. . . . . . . . 9 ⊢ ¬ (𝑥 = 𝑦 ∧ ⊥)
3	2	nex 1803	. . . . . . . 8 ⊢ ¬ ∃𝑥(𝑥 = 𝑦 ∧ ⊥)
4	3	intnan 487	. . . . . . 7 ⊢ ¬ ((𝑥 = 𝑦 → ⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ⊥))
5		dfsb1 2485	. . . . . . 7 ⊢ ([𝑦 / 𝑥]⊥ ↔ ((𝑥 = 𝑦 → ⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ⊥)))
6	4, 5	mtbir 323	. . . . . 6 ⊢ ¬ [𝑦 / 𝑥]⊥
7	6	intnan 487	. . . . 5 ⊢ ¬ (𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥)
8	7	nex 1803	. . . 4 ⊢ ¬ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥)
9	8	intnan 487	. . 3 ⊢ ¬ ((𝑥 = 𝑦 → [𝑦 / 𝑥]⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥))
10		dfsb1 2485	. . 3 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ ↔ ((𝑥 = 𝑦 → [𝑦 / 𝑥]⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥)))
11	9, 10	mtbir 323	. 2 ⊢ ¬ [𝑦 / 𝑥][𝑦 / 𝑥]⊥
12	11	pm2.21i 119	1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∧ wa 396  ⊥wfal 1551  ∃wex 1782  [wsb 2067

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 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068

This theorem is referenced by: (None)