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Theorem bj-ssb1a 32301
Description: One direction of a simplified definition of substitution in case of disjoint variables. See bj-ssb1 32302 for the biconditional, which requires ax-11 2031. (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-ssb1a (∀𝑥(𝑥 = 𝑡𝜑) → [𝑡/𝑥]b𝜑)
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ssb1a
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax-1 6 . . . . . 6 ((𝑥 = 𝑡𝜑) → (∃𝑦 𝑦 = 𝑡 → (𝑥 = 𝑡𝜑)))
2 19.23v 1899 . . . . . 6 (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → (𝑥 = 𝑡𝜑)))
31, 2sylibr 224 . . . . 5 ((𝑥 = 𝑡𝜑) → ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡𝜑)))
4 equequ2 1950 . . . . . . . 8 (𝑦 = 𝑡 → (𝑥 = 𝑦𝑥 = 𝑡))
54imbi1d 331 . . . . . . 7 (𝑦 = 𝑡 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
65pm5.74i 260 . . . . . 6 ((𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 → (𝑥 = 𝑡𝜑)))
76albii 1744 . . . . 5 (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡𝜑)))
83, 7sylibr 224 . . . 4 ((𝑥 = 𝑡𝜑) → ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
98alimi 1736 . . 3 (∀𝑥(𝑥 = 𝑡𝜑) → ∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
10 bj-ssblem2 32300 . . 3 (∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
11 stdpc5v 1864 . . . 4 (∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
1211alimi 1736 . . 3 (∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
139, 10, 123syl 18 . 2 (∀𝑥(𝑥 = 𝑡𝜑) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
14 df-ssb 32289 . 2 ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
1513, 14sylibr 224 1 (∀𝑥(𝑥 = 𝑡𝜑) → [𝑡/𝑥]b𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wex 1701  [wssb 32288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-ssb 32289
This theorem is referenced by: (None)
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