Theorem ax11w 1721

Description: Weak version of ax-11 1746 from which we can prove any ax-11 1746 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). (Contributed by NM, 10-Apr-2017.)

Hypotheses

Ref	Expression
ax11w.1	⊢ (x = y → (φ ↔ ψ))
ax11w.2	⊢ (y = z → (φ ↔ χ))

Assertion

Ref	Expression
ax11w	⊢ (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 ax11w

Step	Hyp	Ref	Expression
1		ax11w.2	. . 3 ⊢ (y = z → (φ ↔ χ))
2	1	spw 1694	. 2 ⊢ (∀yφ → φ)
3		ax11w.1	. . 3 ⊢ (x = y → (φ ↔ ψ))
4	3	ax11wlem 1720	. 2 ⊢ (x = y → (φ → ∀x(x = y → φ)))
5	2, 4	syl5 28	1 ⊢ (x = y → (∀yφ → ∀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  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  ax11wdemo  1723