Theorem mpgbi 989

Description: Modus ponens on biconditional combined with generalization.

Hypotheses

Ref	Expression
mpgbi.1	⊢ (∀xφ ↔ ψ)
mpgbi.2	⊢ φ

Assertion

Ref	Expression
mpgbi	⊢ ψ

Proof of Theorem mpgbi

Step	Hyp	Ref	Expression
1		mpgbi.1	. . 3 ⊢ (∀xφ ↔ ψ)
2	1	biimp 151	. 2 ⊢ (∀xφ → ψ)
3		mpgbi.2	. 2 ⊢ φ
4	2, 3	mpg 988	1 ⊢ ψ

Syntax hints:   ↔ wb 146  ∀wal 956

This theorem is referenced by:  nex 1103  exan 1108  nalset 2717  ac4 4760  ac8 4773  ackm 4792

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965

This theorem depends on definitions:  df-bi 147

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