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Theorem ax12w 2132
Description: Weak version of ax-12 2176 from which we can prove any ax-12 2176 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 2134. (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 2032 . 2 (∀𝑦𝜑𝜑)
3 ax12w.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
43ax12wlem 2131 . 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 206  wal 1537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779
This theorem is referenced by:  ax12wdemo  2134
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