Theorem subsym1 37311

Description: A symmetry with [𝑥 / 𝑦].

See negsym1 37301 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		sbv 2124	. . 3 ⊢ ([𝑦 / 𝑥]⊥ ↔ ⊥)
2		falim 1586	. . 3 ⊢ (⊥ → 𝜑)
3	1, 2	sylbi 220	. 2 ⊢ ([𝑦 / 𝑥]⊥ → 𝜑)
4	3	sbimi 2110	1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4  ⊥wfal 1581  [wsb 2098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099

This theorem is used by: (None)