Theorem sbcimdv 3498

Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1738). (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 3445	. 2 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V)
2		sbcimdv.1	. . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	alrimiv 1855	. . . 4 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
4		spsbc 3448	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒)))
5		sbcim1 3482	. . . 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 1481   ∈ wcel 1990  Vcvv 3200  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  esum2dlem  30154