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 → (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 2041	. 2 ⊢ (∀𝑦𝜑 → 𝜑)
3		ax12w.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	ax12wlem 2132	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	2, 4	syl5 34	1 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

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