Theorem sbcop1 5471

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 3781	. . . . 5 ⊢ ([𝑎 / 𝑥]𝜓 ↔ ∃𝑥(𝑥 = 𝑎 ∧ 𝜓))
2		opeq1 4842	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → ⟨𝑎, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
3	2	equcoms 2047	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ⟨𝑎, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
4	3	eqeq2d 2780	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
5		sbcop.z	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
6	5	biimprd 251	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜓 → 𝜑))
7	4, 6	biimtrdi 256	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜓 → 𝜑)))
8	7	com23 87	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝜓 → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑)))
9	8	imp 411	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝜓) → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
10	9	exlimiv 1957	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝑎 ∧ 𝜓) → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
11	1, 10	sylbi 220	. . . 4 ⊢ ([𝑎 / 𝑥]𝜓 → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
12	11	alrimiv 1954	. . 3 ⊢ ([𝑎 / 𝑥]𝜓 → ∀𝑧(𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
13		opex 5446	. . . 4 ⊢ ⟨𝑎, 𝑦⟩ ∈ V
14	13	sbc6 3784	. . 3 ⊢ ([⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ ∀𝑧(𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
15	12, 14	sylibr 237	. 2 ⊢ ([𝑎 / 𝑥]𝜓 → [⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
16		sbc5 3781	. . 3 ⊢ ([⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ ∃𝑧(𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑))
17	5	biimpd 232	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 → 𝜓))
18	4, 17	biimtrdi 256	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜑 → 𝜓)))
19	18	com3l 90	. . . . . . 7 ⊢ (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜑 → (𝑥 = 𝑎 → 𝜓)))
20	19	imp 411	. . . . . 6 ⊢ ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → (𝑥 = 𝑎 → 𝜓))
21	20	alrimiv 1954	. . . . 5 ⊢ ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → ∀𝑥(𝑥 = 𝑎 → 𝜓))
22		vex 3467	. . . . . 6 ⊢ 𝑎 ∈ V
23	22	sbc6 3784	. . . . 5 ⊢ ([𝑎 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑎 → 𝜓))
24	21, 23	sylibr 237	. . . 4 ⊢ ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → [𝑎 / 𝑥]𝜓)
25	24	exlimiv 1957	. . 3 ⊢ (∃𝑧(𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → [𝑎 / 𝑥]𝜓)
26	16, 25	sylbi 220	. 2 ⊢ ([⟨𝑎, 𝑦⟩ / 𝑧]𝜑 → [𝑎 / 𝑥]𝜓)
27	15, 26	impbii 212	1 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑦⟩ / 𝑧]𝜑)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806  [wsbc 3753  ⟨cop 4600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601

This theorem is referenced by:  sbcop  5472