Theorem subsym1

Description: A symmetry with [ x / y ] .

See negsym1 for more information. (Contributed by Anthony Hart, 11-Sep-2011)

	Ref	Expression
Assertion	subsym1	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑦 / 𝑥 ] ⊥ → [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbv	⊢ ( [ 𝑦 / 𝑥 ] ⊥ ↔ ⊥ )
2		falim	⊢ ( ⊥ → 𝜑 )
3	1 2	sylbi	⊢ ( [ 𝑦 / 𝑥 ] ⊥ → 𝜑 )
4	3	sbimi	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑦 / 𝑥 ] ⊥ → [ 𝑦 / 𝑥 ] 𝜑 )