Theorem or2expropbilem2 47157

Description: Lemma 2 for or2expropbi 47158 and ich2exprop 47595. (Contributed by AV, 16-Jul-2023.)

Assertion

Ref	Expression
or2expropbilem2	⊢ (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑎,𝑏,𝑥,𝑦   𝜑,𝑥,𝑦   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑎,𝑏)

Proof of Theorem or2expropbilem2

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑥(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)
2		nfv 1915	. 2 ⊢ Ⅎ𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)
3		nfv 1915	. . 3 ⊢ Ⅎ𝑎⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩
4		nfcv 2895	. . . 4 ⊢ Ⅎ𝑎𝑦
5		nfsbc1v 3757	. . . 4 ⊢ Ⅎ𝑎[𝑥 / 𝑎]𝜑
6	4, 5	nfsbcw 3759	. . 3 ⊢ Ⅎ𝑎[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
7	3, 6	nfan 1900	. 2 ⊢ Ⅎ𝑎(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
8		nfv 1915	. . 3 ⊢ Ⅎ𝑏⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩
9		nfsbc1v 3757	. . 3 ⊢ Ⅎ𝑏[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
10	8, 9	nfan 1900	. 2 ⊢ Ⅎ𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
11		opeq12 4826	. . . 4 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
12	11	eqeq2d 2744	. . 3 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩))
13		sbceq1a 3748	. . . 4 ⊢ (𝑎 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑎]𝜑))
14		sbceq1a 3748	. . . 4 ⊢ (𝑏 = 𝑦 → ([𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
15	13, 14	sylan9bb 509	. . 3 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
16	12, 15	anbi12d 632	. 2 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
17	1, 2, 7, 10, 16	cbvex2v 2346	1 ⊢ (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780  [wsbc 3737  ⟨cop 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582

This theorem is referenced by:  or2expropbi  47158  ich2exprop  47595