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Theorem ax12w 2124
Description: Weak version of ax-12 2166 from which we can prove any ax-12 2166 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 2126. (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 2033 . 2 (∀𝑦𝜑𝜑)
3 ax12w.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
43ax12wlem 2123 . 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 205  wal 1535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1778
This theorem is referenced by:  ax12wdemo  2126
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