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Theorem bj-ax12 32618
Description: A weaker form of ax-12 2046 and ax12v2 2048, namely the generalization over 𝑥 of the latter. In this statement, all occurrences of 𝑥 are bound. (Contributed by BJ, 26-Dec-2020.)
Assertion
Ref Expression
bj-ax12 𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ax12
StepHypRef Expression
1 ax12v2 2048 . 2 (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
21ax-gen 1721 1 𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704
This theorem is referenced by:  bj-ax12ssb  32619  bj-sb56  32623
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