Theorem sbcimdv 3098

Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1506). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)

Hypothesis

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

Assertion

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

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem sbcimdv

Step	Hyp	Ref	Expression
1		sbcex 3041	. 2 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V)
2		sbcimdv.1	. . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	alrimiv 1922	. . . 4 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
4		spsbc 3044	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒)))
5		sbcim1 3081	. . . 4 ⊢ ([𝐴 / 𝑥](𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
6	3, 4, 5	syl56 34	. . 3 ⊢ (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))
7	6	com3l 81	. 2 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → (𝐴 ∈ V → [𝐴 / 𝑥]𝜒)))
8	1, 7	mpdi 43	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))

Syntax hints:   → wi 4  ∀wal 1396   ∈ wcel 2202  Vcvv 2803  [wsbc 3032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sbc 3033

This theorem is referenced by: (None)