Theorem sbccomlem 3856

Description: Lemma for sbccom 3857. (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 2168	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
2		exdistr 1954	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
3		an12 643	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑦 = 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)))
4	3	exbii 1847	. . . . . 6 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑥(𝑦 = 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)))
5		19.42v 1953	. . . . . 6 ⊢ (∃𝑥(𝑦 = 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝑦 = 𝐵 ∧ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
6	4, 5	bitri 277	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑦 = 𝐵 ∧ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
7	6	exbii 1847	. . . 4 ⊢ (∃𝑦∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑦 = 𝐵 ∧ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
8	1, 2, 7	3bitr3i 303	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑦 = 𝐵 ∧ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
9		sbc5 3803	. . 3 ⊢ ([𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
10		sbc5 3803	. . 3 ⊢ ([𝐵 / 𝑦]∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐵 ∧ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
11	8, 9, 10	3bitr4i 305	. 2 ⊢ ([𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ [𝐵 / 𝑦]∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
12		sbc5 3803	. . 3 ⊢ ([𝐵 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑))
13	12	sbcbii 3832	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]∃𝑦(𝑦 = 𝐵 ∧ 𝜑))
14		sbc5 3803	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
15	14	sbcbii 3832	. 2 ⊢ ([𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑦]∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
16	11, 13, 15	3bitr4i 305	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1536  ∃wex 1779  [wsbc 3775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3499  df-sbc 3776

This theorem is referenced by:  sbccom  3857