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Theorem sbi1OLD 2536
Description: Obsolete version of sbi1 2070 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 2243 . . . . 5 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
2 sbequ2 2243 . . . . 5 (𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → (𝜑𝜓)))
31, 2syl5d 73 . . . 4 (𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑𝜓)))
4 sbequ1 2242 . . . 4 (𝑥 = 𝑦 → (𝜓 → [𝑦 / 𝑥]𝜓))
53, 4syl6d 75 . . 3 (𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)))
65sps 2177 . 2 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)))
7 sb4OLD 2500 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
8 sb4OLD 2500 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))))
9 ax-2 7 . . . . . 6 ((𝑥 = 𝑦 → (𝜑𝜓)) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
109al2imi 1810 . . . . 5 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)))
11 sb2 2498 . . . . 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 184 1 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1529  [wsb 2063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-10 2139  ax-12 2170  ax-13 2384
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1775  df-nf 1779  df-sb 2064
This theorem is referenced by: (None)
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