Theorem sbcori 36267

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

Hypotheses

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

Assertion

Ref	Expression
sbcori	⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ (𝜒 ∨ 𝜂))

Proof of Theorem sbcori

Step	Hyp	Ref	Expression
1		sbcor 3769	. 2 ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓))
2		sbcori.1	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒)
3		sbcori.2	. . 3 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂)
4	2, 3	orbi12i 912	. 2 ⊢ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) ↔ (𝜒 ∨ 𝜂))
5	1, 4	bitri 274	1 ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ (𝜒 ∨ 𝜂))

Syntax hints:   ↔ wb 205   ∨ wo 844  [wsbc 3716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-sbc 3717

This theorem is referenced by: (None)