Theorem sbequi 2059

Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem sbequi

Step	Hyp	Ref	Expression
1		hbsb2 2057	. . . . . 6 ⊢ (¬ ∀z z = x → ([x / z]φ → ∀z[x / z]φ))
2		equvini 1987	. . . . . . . 8 ⊢ (x = y → ∃z(x = z ∧ z = y))
3		stdpc7 1917	. . . . . . . . . 10 ⊢ (x = z → ([x / z]φ → φ))
4		sbequ1 1918	. . . . . . . . . 10 ⊢ (z = y → (φ → [y / z]φ))
5	3, 4	sylan9 638	. . . . . . . . 9 ⊢ ((x = z ∧ z = y) → ([x / z]φ → [y / z]φ))
6	5	eximi 1576	. . . . . . . 8 ⊢ (∃z(x = z ∧ z = y) → ∃z([x / z]φ → [y / z]φ))
7	2, 6	syl 15	. . . . . . 7 ⊢ (x = y → ∃z([x / z]φ → [y / z]φ))
8		19.35 1600	. . . . . . 7 ⊢ (∃z([x / z]φ → [y / z]φ) ↔ (∀z[x / z]φ → ∃z[y / z]φ))
9	7, 8	sylib 188	. . . . . 6 ⊢ (x = y → (∀z[x / z]φ → ∃z[y / z]φ))
10	1, 9	sylan9 638	. . . . 5 ⊢ ((¬ ∀z z = x ∧ x = y) → ([x / z]φ → ∃z[y / z]φ))
11		nfsb2 2058	. . . . . 6 ⊢ (¬ ∀z z = y → Ⅎz[y / z]φ)
12	11	19.9d 1782	. . . . 5 ⊢ (¬ ∀z z = y → (∃z[y / z]φ → [y / z]φ))
13	10, 12	syl9 66	. . . 4 ⊢ ((¬ ∀z z = x ∧ x = y) → (¬ ∀z z = y → ([x / z]φ → [y / z]φ)))
14	13	ex 423	. . 3 ⊢ (¬ ∀z z = x → (x = y → (¬ ∀z z = y → ([x / z]φ → [y / z]φ))))
15	14	com23 72	. 2 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ([x / z]φ → [y / z]φ))))
16		sbequ2 1650	. . . . . 6 ⊢ (z = x → ([x / z]φ → φ))
17	16	sps 1754	. . . . 5 ⊢ (∀z z = x → ([x / z]φ → φ))
18	17	adantr 451	. . . 4 ⊢ ((∀z z = x ∧ x = y) → ([x / z]φ → φ))
19		sbequ1 1918	. . . . 5 ⊢ (x = y → (φ → [y / x]φ))
20		drsb1 2022	. . . . . 6 ⊢ (∀z z = x → ([y / z]φ ↔ [y / x]φ))
21	20	biimprd 214	. . . . 5 ⊢ (∀z z = x → ([y / x]φ → [y / z]φ))
22	19, 21	sylan9r 639	. . . 4 ⊢ ((∀z z = x ∧ x = y) → (φ → [y / z]φ))
23	18, 22	syld 40	. . 3 ⊢ ((∀z z = x ∧ x = y) → ([x / z]φ → [y / z]φ))
24	23	ex 423	. 2 ⊢ (∀z z = x → (x = y → ([x / z]φ → [y / z]φ)))
25		drsb1 2022	. . . . . 6 ⊢ (∀z z = y → ([x / z]φ ↔ [x / y]φ))
26	25	biimpd 198	. . . . 5 ⊢ (∀z z = y → ([x / z]φ → [x / y]φ))
27		stdpc7 1917	. . . . 5 ⊢ (x = y → ([x / y]φ → φ))
28	26, 27	sylan9 638	. . . 4 ⊢ ((∀z z = y ∧ x = y) → ([x / z]φ → φ))
29	4	sps 1754	. . . . 5 ⊢ (∀z z = y → (φ → [y / z]φ))
30	29	adantr 451	. . . 4 ⊢ ((∀z z = y ∧ x = y) → (φ → [y / z]φ))
31	28, 30	syld 40	. . 3 ⊢ ((∀z z = y ∧ x = y) → ([x / z]φ → [y / z]φ))
32	31	ex 423	. 2 ⊢ (∀z z = y → (x = y → ([x / z]φ → [y / z]φ)))
33	15, 24, 32	pm2.61ii 157	1 ⊢ (x = y → ([x / z]φ → [y / z]φ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbequ  2060