Theorem sbcimdv 3766

Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1812). (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 3706	. 2 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V)
2		sbcimdv.1	. . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	alrimiv 1928	. . . 4 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
4		spsbc 3709	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒)))
5		sbcim1 3749	. . . 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 1536   ∈ wcel 2111  Vcvv 3409  [wsbc 3696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-sbc 3697

This theorem is referenced by:  esum2dlem  31579