Theorem sbcani 38132

Description: Distribution of class substitution over conjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)

Hypotheses

Ref	Expression
sbcani.1	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒)
sbcani.2	⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂)

Assertion

Ref	Expression
sbcani	⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜂))

Proof of Theorem sbcani

Step	Hyp	Ref	Expression
1		sbcan 3815	. 2 ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))
2		sbcani.1	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒)
3		sbcani.2	. . 3 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂)
4	2, 3	anbi12i 628	. 2 ⊢ (([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓) ↔ (𝜒 ∧ 𝜂))
5	1, 4	bitri 275	1 ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜂))

Syntax hints:   ↔ wb 206   ∧ wa 395  [wsbc 3765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-sbc 3766

This theorem is referenced by: (None)