Metamath Proof Explorer


Theorem 2sbiev

Description: Conversion of double implicit substitution to explicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See 2sbievw for a version with extra disjoint variables, but based on fewer axioms. (Contributed by AV, 29-Jul-2023) (New usage is discouraged.)

Ref Expression
Hypothesis 2sbiev.1 ( ( 𝑥 = 𝑡𝑦 = 𝑢 ) → ( 𝜑𝜓 ) )
Assertion 2sbiev ( [ 𝑡 / 𝑥 ] [ 𝑢 / 𝑦 ] 𝜑𝜓 )

Proof

Step Hyp Ref Expression
1 2sbiev.1 ( ( 𝑥 = 𝑡𝑦 = 𝑢 ) → ( 𝜑𝜓 ) )
2 nfv 𝑥 𝜓
3 1 sbiedv ( 𝑥 = 𝑡 → ( [ 𝑢 / 𝑦 ] 𝜑𝜓 ) )
4 2 3 sbie ( [ 𝑡 / 𝑥 ] [ 𝑢 / 𝑦 ] 𝜑𝜓 )