Theorem sbcorgOLD 37761

Description: Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcor 3446 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbcorgOLD	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)))

Proof of Theorem sbcorgOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3405	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓)))
2		dfsbcq2 3405	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3		dfsbcq2 3405	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓))
4	2, 3	orbi12d 742	. 2 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)))
5		sbor 2386	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
6	1, 4, 5	vtoclbg 3240	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   = wceq 1475  [wsb 1867   ∈ wcel 1977  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403

This theorem is referenced by:  sbc3orgOLD  37763  sbc3orgVD  38108