Theorem ax12w

Description: Weak version of ax-12 from which we can prove any ax-12 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 ph ). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for ph , see ax12wdemo . (Contributed by NM, 10-Apr-2017)

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

Proof

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