Theorem sbequi 1717

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 1715	. . . 4 ⊢ (∀z z = x ∨ Ⅎz[x / z]φ)
2		nfr 1408	. . . . . 6 ⊢ (Ⅎz[x / z]φ → ([x / z]φ → ∀z[x / z]φ))
3		equvini 1638	. . . . . . 7 ⊢ (x = y → ∃z(x = z ∧ z = y))
4		stdpc7 1650	. . . . . . . . 9 ⊢ (x = z → ([x / z]φ → φ))
5		sbequ1 1648	. . . . . . . . 9 ⊢ (z = y → (φ → [y / z]φ))
6	4, 5	sylan9 389	. . . . . . . 8 ⊢ ((x = z ∧ z = y) → ([x / z]φ → [y / z]φ))
7	6	eximi 1488	. . . . . . 7 ⊢ (∃z(x = z ∧ z = y) → ∃z([x / z]φ → [y / z]φ))
8		19.35-1 1512	. . . . . . 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 66	. . . . 5 ⊢ (Ⅎz[x / z]φ → (x = y → ([x / z]φ → ∃z[y / z]φ)))
11	10	orim2i 677	. . . 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 7	. . 3 ⊢ (∀z z = x ∨ (x = y → ([x / z]φ → ∃z[y / z]φ)))
13		nfsb2or 1715	. . . . 5 ⊢ (∀z z = y ∨ Ⅎz[y / z]φ)
14		19.9t 1530	. . . . . . 7 ⊢ (Ⅎz[y / z]φ → (∃z[y / z]φ ↔ [y / z]φ))
15	14	biimpd 132	. . . . . 6 ⊢ (Ⅎz[y / z]φ → (∃z[y / z]φ → [y / z]φ))
16	15	orim2i 677	. . . . 5 ⊢ ((∀z z = y ∨ Ⅎz[y / z]φ) → (∀z z = y ∨ (∃z[y / z]φ → [y / z]φ)))
17	13, 16	ax-mp 7	. . . 4 ⊢ (∀z z = y ∨ (∃z[y / z]φ → [y / z]φ))
18		ax-1 5	. . . . 5 ⊢ ((∃z[y / z]φ → [y / z]φ) → (x = y → (∃z[y / z]φ → [y / z]φ)))
19	18	orim2i 677	. . . 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 7	. . 3 ⊢ (∀z z = y ∨ (x = y → (∃z[y / z]φ → [y / z]φ)))
21	12, 20	sbequilem 1716	. 2 ⊢ (∀z z = x ∨ (∀z z = y ∨ (x = y → ([x / z]φ → [y / z]φ))))
22		sbequ2 1649	. . . . . . 7 ⊢ (z = x → ([x / z]φ → φ))
23	22	sps 1427	. . . . . 6 ⊢ (∀z z = x → ([x / z]φ → φ))
24	23	adantr 261	. . . . 5 ⊢ ((∀z z = x ∧ x = y) → ([x / z]φ → φ))
25		sbequ1 1648	. . . . . 6 ⊢ (x = y → (φ → [y / x]φ))
26		drsb1 1677	. . . . . . . 8 ⊢ (∀x x = z → ([y / x]φ ↔ [y / z]φ))
27	26	biimpd 132	. . . . . . 7 ⊢ (∀x x = z → ([y / x]φ → [y / z]φ))
28	27	alequcoms 1406	. . . . . 6 ⊢ (∀z z = x → ([y / x]φ → [y / z]φ))
29	25, 28	sylan9r 390	. . . . 5 ⊢ ((∀z z = x ∧ x = y) → (φ → [y / z]φ))
30	24, 29	syld 40	. . . 4 ⊢ ((∀z z = x ∧ x = y) → ([x / z]φ → [y / z]φ))
31	30	ex 108	. . 3 ⊢ (∀z z = x → (x = y → ([x / z]φ → [y / z]φ)))
32		drsb1 1677	. . . . . . . . 9 ⊢ (∀z z = y → ([x / z]φ ↔ [x / y]φ))
33	32	biimpd 132	. . . . . . . 8 ⊢ (∀z z = y → ([x / z]φ → [x / y]φ))
34		stdpc7 1650	. . . . . . . 8 ⊢ (x = y → ([x / y]φ → φ))
35	33, 34	sylan9 389	. . . . . . 7 ⊢ ((∀z z = y ∧ x = y) → ([x / z]φ → φ))
36	5	sps 1427	. . . . . . . 8 ⊢ (∀z z = y → (φ → [y / z]φ))
37	36	adantr 261	. . . . . . 7 ⊢ ((∀z z = y ∧ x = y) → (φ → [y / z]φ))
38	35, 37	syld 40	. . . . . 6 ⊢ ((∀z z = y ∧ x = y) → ([x / z]φ → [y / z]φ))
39	38	ex 108	. . . . 5 ⊢ (∀z z = y → (x = y → ([x / z]φ → [y / z]φ)))
40	39	orim1i 676	. . . 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		pm1.2 672	. . . 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 635	. 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 7	1 ⊢ (x = y → ([x / z]φ → [y / z]φ))

Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 628  ∀wal 1240  Ⅎwnf 1346  ∃wex 1378  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  sbequ  1718