Theorem sbequi 1863

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 1861	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ Ⅎ𝑧[𝑥 / 𝑧]𝜑)
2		nfr 1542	. . . . . 6 ⊢ (Ⅎ𝑧[𝑥 / 𝑧]𝜑 → ([𝑥 / 𝑧]𝜑 → ∀𝑧[𝑥 / 𝑧]𝜑))
3		equvini 1782	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))
4		stdpc7 1794	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ([𝑥 / 𝑧]𝜑 → 𝜑))
5		sbequ1 1792	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝜑 → [𝑦 / 𝑧]𝜑))
6	4, 5	sylan9 409	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
7	6	eximi 1624	. . . . . . 7 ⊢ (∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → ∃𝑧([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
8		19.35-1 1648	. . . . . . 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 763	. . . 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 1861	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑧]𝜑)
14		19.9t 1666	. . . . . . 7 ⊢ (Ⅎ𝑧[𝑦 / 𝑧]𝜑 → (∃𝑧[𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
15	14	biimpd 144	. . . . . 6 ⊢ (Ⅎ𝑧[𝑦 / 𝑧]𝜑 → (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
16	15	orim2i 763	. . . . 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 763	. . . 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 1862	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))))
22		sbequ2 1793	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → 𝜑))
23	22	sps 1561	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → 𝜑))
24	23	adantr 276	. . . . 5 ⊢ ((∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑 → 𝜑))
25		sbequ1 1792	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
26		drsb1 1823	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]𝜑))
27	26	biimpd 144	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑧]𝜑))
28	27	alequcoms 1540	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑧]𝜑))
29	25, 28	sylan9r 410	. . . . 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 115	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
32		drsb1 1823	. . . . . . . . 9 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑥 / 𝑦]𝜑))
33	32	biimpd 144	. . . . . . . 8 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑥 / 𝑦]𝜑))
34		stdpc7 1794	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑))
35	33, 34	sylan9 409	. . . . . . 7 ⊢ ((∀𝑧 𝑧 = 𝑦 ∧ 𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑 → 𝜑))
36	5	sps 1561	. . . . . . . 8 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝜑 → [𝑦 / 𝑧]𝜑))
37	36	adantr 276	. . . . . . 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 115	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
40	39	orim1i 762	. . . 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		pm1.2 758	. . . 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 718	. 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 104   ∨ wo 710  ∀wal 1371  Ⅎwnf 1484  ∃wex 1516  [wsb 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  sbequ  1864