Theorem sbequi 1811

Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)

Assertion

Ref	Expression
sbequi	⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Proof of Theorem sbequi

Step	Hyp	Ref	Expression
1		nfsb2or 1809	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ Ⅎ𝑧[𝑥 / 𝑧]𝜑)
2		nfr 1498	. . . . . 6 ⊢ (Ⅎ𝑧[𝑥 / 𝑧]𝜑 → ([𝑥 / 𝑧]𝜑 → ∀𝑧[𝑥 / 𝑧]𝜑))
3		equvini 1731	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))
4		stdpc7 1743	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ([𝑥 / 𝑧]𝜑 → 𝜑))
5		sbequ1 1741	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝜑 → [𝑦 / 𝑧]𝜑))
6	4, 5	sylan9 406	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
7	6	eximi 1579	. . . . . . 7 ⊢ (∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → ∃𝑧([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
8		19.35-1 1603	. . . . . . 7 ⊢ (∃𝑧([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑) → (∀𝑧[𝑥 / 𝑧]𝜑 → ∃𝑧[𝑦 / 𝑧]𝜑))
9	3, 7, 8	3syl 17	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑧[𝑥 / 𝑧]𝜑 → ∃𝑧[𝑦 / 𝑧]𝜑))
10	2, 9	syl9 72	. . . . 5 ⊢ (Ⅎ𝑧[𝑥 / 𝑧]𝜑 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → ∃𝑧[𝑦 / 𝑧]𝜑)))
11	10	orim2i 750	. . . 4 ⊢ ((∀𝑧 𝑧 = 𝑥 ∨ Ⅎ𝑧[𝑥 / 𝑧]𝜑) → (∀𝑧 𝑧 = 𝑥 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → ∃𝑧[𝑦 / 𝑧]𝜑))))
12	1, 11	ax-mp 5	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → ∃𝑧[𝑦 / 𝑧]𝜑)))
13		nfsb2or 1809	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑧]𝜑)
14		19.9t 1621	. . . . . . 7 ⊢ (Ⅎ𝑧[𝑦 / 𝑧]𝜑 → (∃𝑧[𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
15	14	biimpd 143	. . . . . 6 ⊢ (Ⅎ𝑧[𝑦 / 𝑧]𝜑 → (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
16	15	orim2i 750	. . . . 5 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑧]𝜑) → (∀𝑧 𝑧 = 𝑦 ∨ (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
17	13, 16	ax-mp 5	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 ∨ (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
18		ax-1 6	. . . . 5 ⊢ ((∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑) → (𝑥 = 𝑦 → (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
19	18	orim2i 750	. . . 4 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))))
20	17, 19	ax-mp 5	. . 3 ⊢ (∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
21	12, 20	sbequilem 1810	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))))
22		sbequ2 1742	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → 𝜑))
23	22	sps 1517	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → 𝜑))
24	23	adantr 274	. . . . 5 ⊢ ((∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑 → 𝜑))
25		sbequ1 1741	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
26		drsb1 1771	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]𝜑))
27	26	biimpd 143	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑧]𝜑))
28	27	alequcoms 1496	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑧]𝜑))
29	25, 28	sylan9r 407	. . . . 5 ⊢ ((∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑦) → (𝜑 → [𝑦 / 𝑧]𝜑))
30	24, 29	syld 45	. . . 4 ⊢ ((∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
31	30	ex 114	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
32		drsb1 1771	. . . . . . . . 9 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑥 / 𝑦]𝜑))
33	32	biimpd 143	. . . . . . . 8 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑥 / 𝑦]𝜑))
34		stdpc7 1743	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑))
35	33, 34	sylan9 406	. . . . . . 7 ⊢ ((∀𝑧 𝑧 = 𝑦 ∧ 𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑 → 𝜑))
36	5	sps 1517	. . . . . . . 8 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝜑 → [𝑦 / 𝑧]𝜑))
37	36	adantr 274	. . . . . . 7 ⊢ ((∀𝑧 𝑧 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 → [𝑦 / 𝑧]𝜑))
38	35, 37	syld 45	. . . . . 6 ⊢ ((∀𝑧 𝑧 = 𝑦 ∧ 𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
39	38	ex 114	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
40	39	orim1i 749	. . . 4 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))) → ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))))
41		pm1.2 745	. . . 4 ⊢ (((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))) → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
42	40, 41	syl 14	. . 3 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))) → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
43	31, 42	jaoi 705	. 2 ⊢ ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))) → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
44	21, 43	ax-mp 5	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 697  ∀wal 1329  Ⅎwnf 1436  ∃wex 1468  [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  sbequ  1812