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Theorem bj-substw 34188
 Description: Weak form of the LHS of bj-subst 34187 proved from the core axiom schemes. Compare ax12w 2134. (Contributed by BJ, 26-May-2024.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-substw.is (𝑥 = 𝑡 → (𝜑𝜓))
Assertion
Ref Expression
bj-substw (∃𝑥(𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑡)   𝜓(𝑡)

Proof of Theorem bj-substw
StepHypRef Expression
1 bj-substw.is . . . . 5 (𝑥 = 𝑡 → (𝜑𝜓))
21pm5.32i 578 . . . 4 ((𝑥 = 𝑡𝜑) ↔ (𝑥 = 𝑡𝜓))
32exbii 1849 . . 3 (∃𝑥(𝑥 = 𝑡𝜑) ↔ ∃𝑥(𝑥 = 𝑡𝜓))
4 19.41v 1950 . . 3 (∃𝑥(𝑥 = 𝑡𝜓) ↔ (∃𝑥 𝑥 = 𝑡𝜓))
53, 4bitri 278 . 2 (∃𝑥(𝑥 = 𝑡𝜑) ↔ (∃𝑥 𝑥 = 𝑡𝜓))
61biimprcd 253 . . 3 (𝜓 → (𝑥 = 𝑡𝜑))
76alrimiv 1928 . 2 (𝜓 → ∀𝑥(𝑥 = 𝑡𝜑))
85, 7simplbiim 508 1 (∃𝑥(𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by: (None)
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