Theorem del43b 857

Description: A distinctor elimination lemma for substitution.

Assertion

Ref	Expression
del43b	⊢ (∀x x = y → ([z / x]φ ↔ [z / y]φ))

Proof of Theorem del43b

Step	Hyp	Ref	Expression
1		del43 856	. 2 ⊢ (∀x x = y → ([z / x]φ → [z / y]φ))
2		del43 856	. . 3 ⊢ (∀y y = x → ([z / y]φ → [z / x]φ))
3	2	eq4s 822	. 2 ⊢ (∀x x = y → ([z / y]φ → [z / x]φ))
4	1, 3	impbid 397	1 ⊢ (∀x x = y → ([z / x]φ ↔ [z / y]φ))

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

This theorem is referenced by:  sbco3 915  sbcom 916

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-12 802

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

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