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Theorem bj-ssb1 32758
Description: A simplified definition of substitution in case of disjoint variables. See bj-ssb1a 32757 for the backward implication, which does not require ax-11 2074 (note that here, the version of ax-11 2074 with disjoint setvar metavariables would suffice). Compare sb6 2457. (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-ssb1 ([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ssb1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 19.21v 1908 . . 3 (∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
21albii 1787 . 2 (∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 19.23v 1911 . . . . 5 (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → (𝑥 = 𝑡𝜑)))
4 equequ2 1999 . . . . . . . 8 (𝑦 = 𝑡 → (𝑥 = 𝑦𝑥 = 𝑡))
54imbi1d 330 . . . . . . 7 (𝑦 = 𝑡 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
65pm5.74i 260 . . . . . 6 ((𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 → (𝑥 = 𝑡𝜑)))
76albii 1787 . . . . 5 (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡𝜑)))
8 ax6ev 1947 . . . . . 6 𝑦 𝑦 = 𝑡
98a1bi 351 . . . . 5 ((𝑥 = 𝑡𝜑) ↔ (∃𝑦 𝑦 = 𝑡 → (𝑥 = 𝑡𝜑)))
103, 7, 93bitr4ri 293 . . . 4 ((𝑥 = 𝑡𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
1110albii 1787 . . 3 (∀𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
12 alcom 2077 . . 3 (∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
1311, 12bitri 264 . 2 (∀𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
14 df-ssb 32745 . 2 ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
152, 13, 143bitr4ri 293 1 ([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wex 1744  [wssb 32744
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-11 2074
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-ssb 32745
This theorem is referenced by:  bj-ax12ssb  32760  bj-ssbssblem  32774  bj-ssbcom3lem  32775
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