Theorem sbcori 35402

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 3822	. 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 911	. 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   ∨ wo 843  [wsbc 3772

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3496  df-sbc 3773

This theorem is referenced by: (None)