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Theorem ax12w 2007
Description: Weak version of ax-12 2044 from which we can prove any ax-12 2044 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that 𝑥 and 𝑦 be distinct (unless 𝑥 does not occur in 𝜑). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for 𝜑, see ax12wdemo 2009. (Contributed by NM, 10-Apr-2017.)
Hypotheses
Ref Expression
ax12w.1 (𝑥 = 𝑦 → (𝜑𝜓))
ax12w.2 (𝑦 = 𝑧 → (𝜑𝜒))
Assertion
Ref Expression
ax12w (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable groups:   𝑦,𝑧   𝜓,𝑥   𝜑,𝑧   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧)

Proof of Theorem ax12w
StepHypRef Expression
1 ax12w.2 . . 3 (𝑦 = 𝑧 → (𝜑𝜒))
21spw 1964 . 2 (∀𝑦𝜑𝜑)
3 ax12w.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
43ax12wlem 2006 . 2 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
52, 4syl5 34 1 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478
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
This theorem is referenced by:  ax12wdemo  2009
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