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Theorem sbequi 2466
 Description: An equality theorem for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.)
Assertion
Ref Expression
sbequi (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Proof of Theorem sbequi
StepHypRef Expression
1 equtr 2118 . . 3 (𝑧 = 𝑥 → (𝑥 = 𝑦𝑧 = 𝑦))
2 sbequ2 2064 . . . 4 (𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑𝜑))
3 sbequ1 2277 . . . 4 (𝑧 = 𝑦 → (𝜑 → [𝑦 / 𝑧]𝜑))
42, 3syl9 77 . . 3 (𝑧 = 𝑥 → (𝑧 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
51, 4syld 47 . 2 (𝑧 = 𝑥 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
6 ax13 2355 . . 3 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
7 sp 2215 . . . . . 6 (∀𝑧 𝑧 = 𝑥𝑧 = 𝑥)
87con3i 151 . . . . 5 𝑧 = 𝑥 → ¬ ∀𝑧 𝑧 = 𝑥)
9 sb4 2447 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → ∀𝑧(𝑧 = 𝑥𝜑)))
108, 9syl 17 . . . 4 𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → ∀𝑧(𝑧 = 𝑥𝜑)))
11 equeuclr 2120 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = 𝑦𝑧 = 𝑥))
1211imim1d 82 . . . . . 6 (𝑥 = 𝑦 → ((𝑧 = 𝑥𝜑) → (𝑧 = 𝑦𝜑)))
1312al2imi 1910 . . . . 5 (∀𝑧 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥𝜑) → ∀𝑧(𝑧 = 𝑦𝜑)))
14 sb2 2442 . . . . 5 (∀𝑧(𝑧 = 𝑦𝜑) → [𝑦 / 𝑧]𝜑)
1513, 14syl6 35 . . . 4 (∀𝑧 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥𝜑) → [𝑦 / 𝑧]𝜑))
1610, 15syl9 77 . . 3 𝑧 = 𝑥 → (∀𝑧 𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
176, 16syld 47 . 2 𝑧 = 𝑥 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
185, 17pm2.61i 176 1 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1650  [wsb 2062 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-12 2211  ax-13 2352 This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ex 1875  df-nf 1879  df-sb 2063 This theorem is referenced by:  sbequ  2467
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