Theorem sbcani 35419

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 3816	. 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 277	1 ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜂))

Syntax hints:   ↔ wb 208   ∧ wa 398  [wsbc 3768

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-v 3493  df-sbc 3769

This theorem is referenced by: (None)