Theorem sbccomlem 3799

Description: Lemma for sbccom 3800. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)

Assertion

Ref	Expression
sbccomlem	⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbccomlem

Step	Hyp	Ref	Expression
1		excom 2166	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
2		exdistr 1955	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
3		an12 644	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑦 = 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)))
4	3	exbii 1849	. . . . . 6 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑥(𝑦 = 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)))
5		19.42v 1954	. . . . . 6 ⊢ (∃𝑥(𝑦 = 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝑦 = 𝐵 ∧ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
6	4, 5	bitri 278	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑦 = 𝐵 ∧ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
7	6	exbii 1849	. . . 4 ⊢ (∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑦 = 𝐵 ∧ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
8	1, 2, 7	3bitr3i 304	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑦 = 𝐵 ∧ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
9		sbc5 3748	. . 3 ⊢ ([𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
10		sbc5 3748	. . 3 ⊢ ([𝐵 / 𝑦]∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐵 ∧ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
11	8, 9, 10	3bitr4i 306	. 2 ⊢ ([𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ [𝐵 / 𝑦]∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
12		sbc5 3748	. . 3 ⊢ ([𝐵 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑))
13	12	sbcbii 3776	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑))
14		sbc5 3748	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
15	14	sbcbii 3776	. 2 ⊢ ([𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑦]∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
16	11, 13, 15	3bitr4i 306	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781  [wsbc 3720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721

This theorem is referenced by:  sbccom  3800