Theorem sbcimdvOLD 3757

Description: Obsolete version of sbcimdv 3756 as of 12-Oct-2024. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sbcimdvOLD.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		sbcex 3693	. 2 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V)
2		sbcimdvOLD.1	. . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	alrimiv 1935	. . . 4 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
4		spsbc 3696	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒)))
5		sbcim1 3739	. . . 4 ⊢ ([𝐴 / 𝑥](𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
6	3, 4, 5	syl56 36	. . 3 ⊢ (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))
7	6	com3l 89	. 2 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → (𝐴 ∈ V → [𝐴 / 𝑥]𝜒)))
8	1, 7	mpdi 45	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))

Syntax hints:   → wi 4  ∀wal 1541   ∈ wcel 2112  Vcvv 3398  [wsbc 3683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-v 3400  df-sbc 3684

This theorem is referenced by: (None)