Theorem sbcimdv 3786

Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1814). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)

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

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sbc 3712	. . . 4 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
2		dfclel 2818	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜓}))
3		df-clab 2716	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
4	3	anbi2i 622	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ (𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜓))
5	4	exbii 1851	. . . 4 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜓))
6	1, 2, 5	3bitri 296	. . 3 ⊢ ([𝐴 / 𝑥]𝜓 ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜓))
7	6	biimpi 215	. 2 ⊢ ([𝐴 / 𝑥]𝜓 → ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜓))
8		sbcimdv.1	. . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒))
9	8	sbimdv 2082	. . . 4 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
10	9	anim2d 611	. . 3 ⊢ (𝜑 → ((𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜓) → (𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜒)))
11	10	eximdv 1921	. 2 ⊢ (𝜑 → (∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜓) → ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜒)))
12		df-sbc 3712	. . . 4 ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝐴 ∈ {𝑥 ∣ 𝜒})
13		dfclel 2818	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜒} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜒}))
14		df-clab 2716	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜒} ↔ [𝑦 / 𝑥]𝜒)
15	14	anbi2i 622	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜒}) ↔ (𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜒))
16	15	exbii 1851	. . . 4 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜒}) ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜒))
17	12, 13, 16	3bitrri 297	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜒) ↔ [𝐴 / 𝑥]𝜒)
18	17	biimpi 215	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜒) → [𝐴 / 𝑥]𝜒)
19	7, 11, 18	syl56 36	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783  [wsb 2068   ∈ wcel 2108  {cab 2715  [wsbc 3711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-sb 2069  df-clab 2716  df-clel 2817  df-sbc 3712

This theorem is referenced by:  esum2dlem  31960