Theorem sbcop1 5508

Description: The proper substitution of an ordered pair for a setvar variable corresponds to a proper substitution of its first component. (Contributed by AV, 8-Apr-2023.)

Hypothesis

Ref	Expression
sbcop.z	⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcop1	⊢ ([𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑦⟩ / 𝑧]𝜑)

Distinct variable groups:   𝑥,𝑎,𝑦,𝑧   𝜑,𝑥,𝑦   𝜓,𝑧

Allowed substitution hints:   𝜑(𝑧,𝑎)   𝜓(𝑥,𝑦,𝑎)

Proof of Theorem sbcop1

Step	Hyp	Ref	Expression
1		sbc5 3832	. . . . 5 ⊢ ([𝑎 / 𝑥]𝜓 ↔ ∃𝑥(𝑥 = 𝑎 ∧ 𝜓))
2		opeq1 4897	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → ⟨𝑎, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
3	2	equcoms 2019	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ⟨𝑎, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
4	3	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
5		sbcop.z	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
6	5	biimprd 248	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜓 → 𝜑))
7	4, 6	biimtrdi 253	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜓 → 𝜑)))
8	7	com23 86	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝜓 → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑)))
9	8	imp 406	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝜓) → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
10	9	exlimiv 1929	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝑎 ∧ 𝜓) → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
11	1, 10	sylbi 217	. . . 4 ⊢ ([𝑎 / 𝑥]𝜓 → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
12	11	alrimiv 1926	. . 3 ⊢ ([𝑎 / 𝑥]𝜓 → ∀𝑧(𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
13		opex 5484	. . . 4 ⊢ ⟨𝑎, 𝑦⟩ ∈ V
14	13	sbc6 3836	. . 3 ⊢ ([⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ ∀𝑧(𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
15	12, 14	sylibr 234	. 2 ⊢ ([𝑎 / 𝑥]𝜓 → [⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
16		sbc5 3832	. . 3 ⊢ ([⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ ∃𝑧(𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑))
17	5	biimpd 229	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 → 𝜓))
18	4, 17	biimtrdi 253	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜑 → 𝜓)))
19	18	com3l 89	. . . . . . 7 ⊢ (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜑 → (𝑥 = 𝑎 → 𝜓)))
20	19	imp 406	. . . . . 6 ⊢ ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → (𝑥 = 𝑎 → 𝜓))
21	20	alrimiv 1926	. . . . 5 ⊢ ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → ∀𝑥(𝑥 = 𝑎 → 𝜓))
22		vex 3492	. . . . . 6 ⊢ 𝑎 ∈ V
23	22	sbc6 3836	. . . . 5 ⊢ ([𝑎 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑎 → 𝜓))
24	21, 23	sylibr 234	. . . 4 ⊢ ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → [𝑎 / 𝑥]𝜓)
25	24	exlimiv 1929	. . 3 ⊢ (∃𝑧(𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → [𝑎 / 𝑥]𝜓)
26	16, 25	sylbi 217	. 2 ⊢ ([⟨𝑎, 𝑦⟩ / 𝑧]𝜑 → [𝑎 / 𝑥]𝜓)
27	15, 26	impbii 209	1 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑦⟩ / 𝑧]𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777  [wsbc 3804  ⟨cop 4654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655

This theorem is referenced by:  sbcop  5509