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Theorem ax12w 1958
Description: Weak version of ax-12 1983 from which we can prove any ax-12 1983 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 1960. (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 1916 . 2 (∀𝑦𝜑𝜑)
3 ax12w.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
43ax12wlem 1957 . 2 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
52, 4syl5 33 1 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  ax12wdemo  1960
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