Theorem subsym1 36467

Description: A symmetry with [𝑥 / 𝑦].

See negsym1 36457 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 2091	. . 3 ⊢ ([𝑦 / 𝑥]⊥ ↔ ⊥)
2		falim 1558	. . 3 ⊢ (⊥ → 𝜑)
3	1, 2	sylbi 217	. 2 ⊢ ([𝑦 / 𝑥]⊥ → 𝜑)
4	3	sbimi 2077	1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ⊥wfal 1553  [wsb 2067

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 1968

This theorem depends on definitions:  df-bi 207  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068

This theorem is referenced by: (None)