Theorem ddelimf 908

Description: Version of ddelim 1000 without any variable restrictions.

Hypotheses

Ref	Expression
ddelimf.1	⊢ (φ → ∀xφ)
ddelimf.2	⊢ (ψ → ∀zψ)
ddelimf.3	⊢ (z = y → (φ ↔ ψ))

Assertion

Ref	Expression
ddelimf	⊢ (¬ ∀x x = y → (ψ → ∀xψ))

Proof of Theorem ddelimf

Step	Hyp	Ref	Expression
1		ddelimf.1	. . 3 ⊢ (φ → ∀xφ)
2	1	hbsb4 905	. 2 ⊢ (¬ ∀x x = y → ([y / z]φ → ∀x[y / z]φ))
3		ddelimf.2	. . 3 ⊢ (ψ → ∀zψ)
4		ddelimf.3	. . 3 ⊢ (z = y → (φ ↔ ψ))
5	3, 4	sbie 904	. 2 ⊢ ([y / z]φ ↔ ψ)
6	5	bial 695	. 2 ⊢ (∀x[y / z]φ ↔ ∀xψ)
7	2, 5, 6	3imtr3g 425	1 ⊢ (¬ ∀x x = y → (ψ → ∀xψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672   = weq 797  [wsb 852

This theorem is referenced by:  ddelim 1000

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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