Theorem sbcimdvOLD 3466

Description: Obsolete proof of sbcimdv 3465 as of 7-Jul-2021. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
sbcimdv.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
sbcimdvOLD	⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem sbcimdvOLD

Step	Hyp	Ref	Expression
1		sbcimdv.1	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	alrimiv 1842	. . 3 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
3		spsbc 3415	. . 3 ⊢ (𝐴 ∈ V → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒)))
4		sbcim1 3449	. . 3 ⊢ ([𝐴 / 𝑥](𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
5	2, 3, 4	syl56 35	. 2 ⊢ (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))
6		sbcex 3412	. . . . 5 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V)
7	6	con3i 149	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜓)
8	7	pm2.21d 117	. . 3 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
9	8	a1d 25	. 2 ⊢ (¬ 𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))
10	5, 9	pm2.61i 175	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473   ∈ wcel 1977  Vcvv 3173  [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: (None)