Theorem 2sbbii 2114

Description: Infer double substitution into both sides of a logical equivalence. (Contributed by AV, 30-Jul-2023.)

Hypothesis

Ref	Expression
sbbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
2sbbii	⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)

Proof of Theorem 2sbbii

Step	Hyp	Ref	Expression
1		sbbii.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	sbbii 2113	. 2 ⊢ ([𝑢 / 𝑦]𝜑 ↔ [𝑢 / 𝑦]𝜓)
3	2	sbbii 2113	1 ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)

This proof depends on syntax axioms:   ↔ wb 209  [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-ex 1813  df-sb 2100

This theorem is used by:  sbco4lemOLD  2211  ichcircshi  48244  ichbi12i  48250  icheq  48252  ichal  48256