Theorem wl-sbcom2d 32523

Description: Version of sbcom2 2433 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)

Hypotheses

Ref	Expression
wl-sbcom2d.1	⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑤)
wl-sbcom2d.2	⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)
wl-sbcom2d.3	⊢ (𝜑 → ¬ ∀𝑧 𝑧 = 𝑦)

Assertion

Ref	Expression
wl-sbcom2d	⊢ (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))

Proof of Theorem wl-sbcom2d

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6ev 1877	. 2 ⊢ ∃𝑢 𝑢 = 𝑦
2		ax6ev 1877	. 2 ⊢ ∃𝑣 𝑣 = 𝑤
3		wl-sbcom2d.2	. . . . . . . 8 ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)
4		wl-sbcom2d-lem2 32522	. . . . . . . . . . 11 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜓 ↔ ∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜓)))
5		alcom 2024	. . . . . . . . . . . 12 ⊢ (∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜓) ↔ ∀𝑧∀𝑥((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜓))
6		ancomst 467	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜓) ↔ ((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜓))
7	6	2albii 1738	. . . . . . . . . . . 12 ⊢ (∀𝑧∀𝑥((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜓) ↔ ∀𝑧∀𝑥((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜓))
8	5, 7	bitri 263	. . . . . . . . . . 11 ⊢ (∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜓) ↔ ∀𝑧∀𝑥((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜓))
9	4, 8	syl6bb 275	. . . . . . . . . 10 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜓 ↔ ∀𝑧∀𝑥((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜓)))
10	9	naecoms 2301	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜓 ↔ ∀𝑧∀𝑥((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜓)))
11		wl-sbcom2d-lem2 32522	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑣 / 𝑧][𝑢 / 𝑥]𝜓 ↔ ∀𝑧∀𝑥((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜓)))
12	10, 11	bitr4d 270	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜓 ↔ [𝑣 / 𝑧][𝑢 / 𝑥]𝜓))
13	3, 12	syl 17	. . . . . . 7 ⊢ (𝜑 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜓 ↔ [𝑣 / 𝑧][𝑢 / 𝑥]𝜓))
14	13	adantl 481	. . . . . 6 ⊢ (((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) ∧ 𝜑) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜓 ↔ [𝑣 / 𝑧][𝑢 / 𝑥]𝜓))
15		wl-sbcom2d.1	. . . . . . . 8 ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑤)
16		wl-sbcom2d-lem1 32521	. . . . . . . 8 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓)))
17	15, 16	syl5 33	. . . . . . 7 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → (𝜑 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓)))
18	17	imp 444	. . . . . 6 ⊢ (((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) ∧ 𝜑) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))
19		wl-sbcom2d.3	. . . . . . . . 9 ⊢ (𝜑 → ¬ ∀𝑧 𝑧 = 𝑦)
20		wl-sbcom2d-lem1 32521	. . . . . . . . 9 ⊢ ((𝑣 = 𝑤 ∧ 𝑢 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑣 / 𝑧][𝑢 / 𝑥]𝜓 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜓)))
21	19, 20	syl5 33	. . . . . . . 8 ⊢ ((𝑣 = 𝑤 ∧ 𝑢 = 𝑦) → (𝜑 → ([𝑣 / 𝑧][𝑢 / 𝑥]𝜓 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜓)))
22	21	ancoms 468	. . . . . . 7 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → (𝜑 → ([𝑣 / 𝑧][𝑢 / 𝑥]𝜓 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜓)))
23	22	imp 444	. . . . . 6 ⊢ (((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) ∧ 𝜑) → ([𝑣 / 𝑧][𝑢 / 𝑥]𝜓 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜓))
24	14, 18, 23	3bitr3rd 298	. . . . 5 ⊢ (((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) ∧ 𝜑) → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))
25	24	exp31 628	. . . 4 ⊢ (𝑢 = 𝑦 → (𝑣 = 𝑤 → (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))))
26	25	exlimdv 1848	. . 3 ⊢ (𝑢 = 𝑦 → (∃𝑣 𝑣 = 𝑤 → (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))))
27	26	exlimiv 1845	. 2 ⊢ (∃𝑢 𝑢 = 𝑦 → (∃𝑣 𝑣 = 𝑤 → (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))))
28	1, 2, 27	mp2 9	1 ⊢ (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868

This theorem is referenced by: (None)