Theorem sbccom2lem 37455

Description: Lemma for sbccom2 37456. (Contributed by Giovanni Mascellani, 31-May-2019.)

Hypothesis

Ref	Expression
sbccom2lem.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
sbccom2lem	⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem sbccom2lem

Step	Hyp	Ref	Expression
1		sbcan 3829	. . . 4 ⊢ ([𝐴 / 𝑥](𝑦 = 𝐵 ∧ 𝜑) ↔ ([𝐴 / 𝑥]𝑦 = 𝐵 ∧ [𝐴 / 𝑥]𝜑))
2		sbc5 3805	. . . 4 ⊢ ([𝐴 / 𝑥](𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
3		sbccom2lem.1	. . . . . 6 ⊢ 𝐴 ∈ V
4	3	csbconstgi 3915	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝑦 = 𝑦
5		eqid 2731	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
6	3, 4, 5	sbceqi 4410	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)
7	6	anbi1i 623	. . . 4 ⊢ (([𝐴 / 𝑥]𝑦 = 𝐵 ∧ [𝐴 / 𝑥]𝜑) ↔ (𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
8	1, 2, 7	3bitr3i 301	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
9	8	exbii 1849	. 2 ⊢ (∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
10		sbc5 3805	. . . . 5 ⊢ ([𝐵 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑))
11	10	sbcbii 3837	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑))
12		sbc5 3805	. . . 4 ⊢ ([𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
13	11, 12	bitri 275	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
14		19.42v 1956	. . . . . 6 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
15	14	bicomi 223	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
16	15	exbii 1849	. . . 4 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
17		excom 2161	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
18	16, 17	bitri 275	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
19	13, 18	bitri 275	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
20		sbc5 3805	. 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
21	9, 19, 20	3bitr4i 303	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1540  ∃wex 1780   ∈ wcel 2105  Vcvv 3473  [wsbc 3777  ⦋csb 3893

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475  df-sbc 3778  df-csb 3894

This theorem is referenced by:  sbccom2  37456