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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
StepHypRef Expression
1 hbsb2 2057 . . . . . 6 z z = x → ([x / z]φz[x / z]φ))
2 equvini 1987 . . . . . . . 8 (x = yz(x = z z = y))
3 stdpc7 1917 . . . . . . . . . 10 (x = z → ([x / z]φφ))
4 sbequ1 1918 . . . . . . . . . 10 (z = y → (φ → [y / z]φ))
53, 4sylan9 638 . . . . . . . . 9 ((x = z z = y) → ([x / z]φ → [y / z]φ))
65eximi 1576 . . . . . . . 8 (z(x = z z = y) → z([x / z]φ → [y / z]φ))
72, 6syl 15 . . . . . . 7 (x = yz([x / z]φ → [y / z]φ))
8 19.35 1600 . . . . . . 7 (z([x / z]φ → [y / z]φ) ↔ (z[x / z]φz[y / z]φ))
97, 8sylib 188 . . . . . 6 (x = y → (z[x / z]φz[y / z]φ))
101, 9sylan9 638 . . . . 5 ((¬ z z = x x = y) → ([x / z]φz[y / z]φ))
11 nfsb2 2058 . . . . . 6 z z = y → Ⅎz[y / z]φ)
121119.9d 1782 . . . . 5 z z = y → (z[y / z]φ → [y / z]φ))
1310, 12syl9 66 . . . 4 ((¬ z z = x x = y) → (¬ z z = y → ([x / z]φ → [y / z]φ)))
1413ex 423 . . 3 z z = x → (x = y → (¬ z z = y → ([x / z]φ → [y / z]φ))))
1514com23 72 . 2 z z = x → (¬ z z = y → (x = y → ([x / z]φ → [y / z]φ))))
16 sbequ2 1650 . . . . . 6 (z = x → ([x / z]φφ))
1716sps 1754 . . . . 5 (z z = x → ([x / z]φφ))
1817adantr 451 . . . 4 ((z z = x x = y) → ([x / z]φφ))
19 sbequ1 1918 . . . . 5 (x = y → (φ → [y / x]φ))
20 drsb1 2022 . . . . . 6 (z z = x → ([y / z]φ ↔ [y / x]φ))
2120biimprd 214 . . . . 5 (z z = x → ([y / x]φ → [y / z]φ))
2219, 21sylan9r 639 . . . 4 ((z z = x x = y) → (φ → [y / z]φ))
2318, 22syld 40 . . 3 ((z z = x x = y) → ([x / z]φ → [y / z]φ))
2423ex 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))
2625biimpd 198 . . . . 5 (z z = y → ([x / z]φ → [x / y]φ))
27 stdpc7 1917 . . . . 5 (x = y → ([x / y]φφ))
2826, 27sylan9 638 . . . 4 ((z z = y x = y) → ([x / z]φφ))
294sps 1754 . . . . 5 (z z = y → (φ → [y / z]φ))
3029adantr 451 . . . 4 ((z z = y x = y) → (φ → [y / z]φ))
3128, 30syld 40 . . 3 ((z z = y x = y) → ([x / z]φ → [y / z]φ))
3231ex 423 . 2 (z z = y → (x = y → ([x / z]φ → [y / z]φ)))
3315, 24, 32pm2.61ii 157 1 (x = y → ([x / z]φ → [y / z]φ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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