Theorem sbccom2lem 38498

Description: Lemma for sbccom2 38499. (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 3779	. . . 4 ⊢ ([𝐴 / 𝑥](𝑦 = 𝐵 ∧ 𝜑) ↔ ([𝐴 / 𝑥]𝑦 = 𝐵 ∧ [𝐴 / 𝑥]𝜑))
2		sbc5 3758	. . . 4 ⊢ ([𝐴 / 𝑥](𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
3		sbccom2lem.1	. . . . . 6 ⊢ 𝐴 ∈ V
4	3	csbconstgi 3859	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝑦 = 𝑦
5		eqid 2740	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
6	3, 4, 5	sbceqi 4348	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)
7	6	anbi1i 630	. . . 4 ⊢ (([𝐴 / 𝑥]𝑦 = 𝐵 ∧ [𝐴 / 𝑥]𝜑) ↔ (𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
8	1, 2, 7	3bitr3i 302	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
9	8	exbii 1855	. 2 ⊢ (∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
10		sbc5 3758	. . . . 5 ⊢ ([𝐵 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑))
11	10	sbcbii 3786	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑))
12		sbc5 3758	. . . 4 ⊢ ([𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
13	11, 12	bitri 276	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
14		19.42v 1960	. . . . . 6 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
15	14	bicomi 225	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
16	15	exbii 1855	. . . 4 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
17		excom 2173	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
18	16, 17	bitri 276	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
19	13, 18	bitri 276	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
20		sbc5 3758	. 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
21	9, 19, 20	3bitr4i 304	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3432  [wsbc 3730  ⦋csb 3838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-v 3434  df-sbc 3731  df-csb 3839

This theorem is referenced by:  sbccom2  38499