Theorem sbccom2lem 39259

Description: Lemma for sbccom2 39260. (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 3791	. . . 4 ⊢ ([𝐴 / 𝑥](𝑦 = 𝐵 ∧ 𝜑) ↔ ([𝐴 / 𝑥]𝑦 = 𝐵 ∧ [𝐴 / 𝑥]𝜑))
2		sbc5 3770	. . . 4 ⊢ ([𝐴 / 𝑥](𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
3		sbccom2lem.1	. . . . . 6 ⊢ 𝐴 ∈ V
4	3	csbconstgi 3872	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝑦 = 𝑦
5		eqid 2763	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
6	3, 4, 5	sbceqi 4374	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)
7	6	anbi1i 636	. . . 4 ⊢ (([𝐴 / 𝑥]𝑦 = 𝐵 ∧ [𝐴 / 𝑥]𝜑) ↔ (𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
8	1, 2, 7	3bitr3i 304	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
9	8	exbii 1881	. 2 ⊢ (∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
10		sbc5 3770	. . . . 5 ⊢ ([𝐵 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑))
11	10	sbcbii 3798	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑))
12		sbc5 3770	. . . 4 ⊢ ([𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
13	11, 12	bitri 278	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
14		19.42v 1988	. . . . . 6 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
15	14	bicomi 227	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
16	15	exbii 1881	. . . 4 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
17		excom 2202	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
18	16, 17	bitri 278	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
19	13, 18	bitri 278	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
20		sbc5 3770	. 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ∧ [𝐴 / 𝑥]𝜑))
21	9, 19, 20	3bitr4i 306	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454  [wsbc 3742  ⦋csb 3851

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-v 3456  df-sbc 3743  df-csb 3852

This theorem is used by:  sbccom2  39260