Theorem ax12w 2174

Description: Weak version of ax-12 2219 from which we can prove any ax-12 2219 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 2176. (Contributed by NM, 10-Apr-2017.)

Hypotheses

Ref	Expression
ax12w.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
ax12w.2	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
ax12w	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

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

Step	Hyp	Ref	Expression
1		ax12w.2	. . 3 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))
2	1	spw 2061	. 2 ⊢ (∀𝑦𝜑 → 𝜑)
3		ax12w.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	ax12wlem 2173	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	2, 4	syl5 35	1 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807

This theorem is referenced by:  ax12wdemo  2176