Theorem sbcani 38158

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 3786	. 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 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-sbc 3737

This theorem is referenced by: (None)