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Theorem ax12w 2133
Description: Weak version of ax-12 2175 from which we can prove any ax-12 2175 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 2135. (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 2041 . 2 (∀𝑦𝜑𝜑)
3 ax12w.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
43ax12wlem 2132 . 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 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787
This theorem is referenced by:  ax12wdemo  2135
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