Theorem subsym1 33777

Description: A symmetry with [𝑥 / 𝑦].

See negsym1 33767 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 1551	. . . . . . . . . 10 ⊢ ¬ ⊥
2	1	intnan 489	. . . . . . . . 9 ⊢ ¬ (𝑥 = 𝑦 ∧ ⊥)
3	2	nex 1801	. . . . . . . 8 ⊢ ¬ ∃𝑥(𝑥 = 𝑦 ∧ ⊥)
4	3	intnan 489	. . . . . . 7 ⊢ ¬ ((𝑥 = 𝑦 → ⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ⊥))
5		dfsb1 2510	. . . . . . 7 ⊢ ([𝑦 / 𝑥]⊥ ↔ ((𝑥 = 𝑦 → ⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ⊥)))
6	4, 5	mtbir 325	. . . . . 6 ⊢ ¬ [𝑦 / 𝑥]⊥
7	6	intnan 489	. . . . 5 ⊢ ¬ (𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥)
8	7	nex 1801	. . . 4 ⊢ ¬ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥)
9	8	intnan 489	. . 3 ⊢ ¬ ((𝑥 = 𝑦 → [𝑦 / 𝑥]⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥))
10		dfsb1 2510	. . 3 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ ↔ ((𝑥 = 𝑦 → [𝑦 / 𝑥]⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥)))
11	9, 10	mtbir 325	. 2 ⊢ ¬ [𝑦 / 𝑥][𝑦 / 𝑥]⊥
12	11	pm2.21i 119	1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∧ wa 398  ⊥wfal 1549  ∃wex 1780  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by: (None)