Theorem ax11i 1647

Description: Inference that has ax-11 1746 (without ∀y) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)

Hypotheses

Ref	Expression
ax11i.1	⊢ (x = y → (φ ↔ ψ))
ax11i.2	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
ax11i	⊢ (x = y → (φ → ∀x(x = y → φ)))

Proof of Theorem ax11i

Step	Hyp	Ref	Expression
1		ax11i.1	. 2 ⊢ (x = y → (φ ↔ ψ))
2		ax11i.2	. . 3 ⊢ (ψ → ∀xψ)
3	1	biimprcd 216	. . 3 ⊢ (ψ → (x = y → φ))
4	2, 3	alrimih 1565	. 2 ⊢ (ψ → ∀x(x = y → φ))
5	1, 4	syl6bi 219	1 ⊢ (x = y → (φ → ∀x(x = y → φ)))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  ax11wlem  1720