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Theorem sbi1OLD 2520
 Description: Obsolete version of sbi1 2077 as of 24-Jul-2023. Removal of implication from substitution. (Contributed by NM, 14-May-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbi1OLD ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem sbi1OLD
StepHypRef Expression
1 sbequ2 2248 . . . . 5 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
2 sbequ2 2248 . . . . 5 (𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → (𝜑𝜓)))
31, 2syl5d 73 . . . 4 (𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑𝜓)))
4 sbequ1 2247 . . . 4 (𝑥 = 𝑦 → (𝜓 → [𝑦 / 𝑥]𝜓))
53, 4syl6d 75 . . 3 (𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)))
65sps 2183 . 2 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)))
7 sb4OLD 2496 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
8 sb4OLD 2496 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))))
9 ax-2 7 . . . . . 6 ((𝑥 = 𝑦 → (𝜑𝜓)) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
109al2imi 1818 . . . . 5 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)))
11 sb2 2494 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜓) → [𝑦 / 𝑥]𝜓)
1210, 11syl6 35 . . . 4 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜓))
138, 12syl6 35 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜓)))
147, 13syl5d 73 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)))
156, 14pm2.61i 185 1 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-ex 1783  df-nf 1787  df-sb 2071 This theorem is referenced by: (None)
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