Theorem 2sbbii 2082

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 2081	. 2 ⊢ ([𝑢 / 𝑦]𝜑 ↔ [𝑢 / 𝑦]𝜓)
3	2	sbbii 2081	1 ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)

Syntax hints:   ↔ wb 208  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-sb 2070

This theorem is referenced by:  sbco4lem  2283  ichcircshi  43661  ichbi12i  43667  icheq  43669  ichal  43676