Theorem subsym1 36995

Description: A symmetry with [𝑥 / 𝑦].

See negsym1 36985 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 2125	. . 3 ⊢ ([𝑦 / 𝑥]⊥ ↔ ⊥)
2		falim 1587	. . 3 ⊢ (⊥ → 𝜑)
3	1, 2	sylbi 220	. 2 ⊢ ([𝑦 / 𝑥]⊥ → 𝜑)
4	3	sbimi 2111	1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4  ⊥wfal 1582  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100

This theorem is used by: (None)