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Theorem sbequi 2403
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 1994 . . 3 (𝑧 = 𝑥 → (𝑥 = 𝑦𝑧 = 𝑦))
2 sbequ2 1939 . . . 4 (𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑𝜑))
3 sbequ1 2148 . . . 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 2285 . . 3 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
7 sp 2091 . . . . . 6 (∀𝑧 𝑧 = 𝑥𝑧 = 𝑥)
87con3i 150 . . . . 5 𝑧 = 𝑥 → ¬ ∀𝑧 𝑧 = 𝑥)
9 sb4 2384 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → ∀𝑧(𝑧 = 𝑥𝜑)))
108, 9syl 17 . . . 4 𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → ∀𝑧(𝑧 = 𝑥𝜑)))
11 equeuclr 1996 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = 𝑦𝑧 = 𝑥))
1211imim1d 82 . . . . . 6 (𝑥 = 𝑦 → ((𝑧 = 𝑥𝜑) → (𝑧 = 𝑦𝜑)))
1312al2imi 1783 . . . . 5 (∀𝑧 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥𝜑) → ∀𝑧(𝑧 = 𝑦𝜑)))
14 sb2 2380 . . . . 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 1521  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  sbequ  2404
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