Theorem sbcop1 5450

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 3767	. . . . 5 ⊢ ([𝑎 / 𝑥]𝜓 ↔ ∃𝑥(𝑥 = 𝑎 ∧ 𝜓))
2		opeq1 4825	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → ⟨𝑎, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
3	2	equcoms 2034	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ⟨𝑎, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
4	3	eqeq2d 2767	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
5		sbcop.z	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
6	5	biimprd 250	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜓 → 𝜑))
7	4, 6	biimtrdi 255	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜓 → 𝜑)))
8	7	com23 86	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝜓 → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑)))
9	8	imp 409	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝜓) → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
10	9	exlimiv 1944	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝑎 ∧ 𝜓) → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
11	1, 10	sylbi 219	. . . 4 ⊢ ([𝑎 / 𝑥]𝜓 → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
12	11	alrimiv 1941	. . 3 ⊢ ([𝑎 / 𝑥]𝜓 → ∀𝑧(𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
13		opex 5425	. . . 4 ⊢ ⟨𝑎, 𝑦⟩ ∈ V
14	13	sbc6 3770	. . 3 ⊢ ([⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ ∀𝑧(𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
15	12, 14	sylibr 236	. 2 ⊢ ([𝑎 / 𝑥]𝜓 → [⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
16		sbc5 3767	. . 3 ⊢ ([⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ ∃𝑧(𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑))
17	5	biimpd 231	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 → 𝜓))
18	4, 17	biimtrdi 255	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜑 → 𝜓)))
19	18	com3l 89	. . . . . . 7 ⊢ (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜑 → (𝑥 = 𝑎 → 𝜓)))
20	19	imp 409	. . . . . 6 ⊢ ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → (𝑥 = 𝑎 → 𝜓))
21	20	alrimiv 1941	. . . . 5 ⊢ ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → ∀𝑥(𝑥 = 𝑎 → 𝜓))
22		vex 3452	. . . . . 6 ⊢ 𝑎 ∈ V
23	22	sbc6 3770	. . . . 5 ⊢ ([𝑎 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑎 → 𝜓))
24	21, 23	sylibr 236	. . . 4 ⊢ ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → [𝑎 / 𝑥]𝜓)
25	24	exlimiv 1944	. . 3 ⊢ (∃𝑧(𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → [𝑎 / 𝑥]𝜓)
26	16, 25	sylbi 219	. 2 ⊢ ([⟨𝑎, 𝑦⟩ / 𝑧]𝜑 → [𝑎 / 𝑥]𝜓)
27	15, 26	impbii 211	1 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑦⟩ / 𝑧]𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1552   = wceq 1554  ∃wex 1793  [wsbc 3739  ⟨cop 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-rab 3409  df-v 3450  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583

This theorem is referenced by:  sbcop  5451