Theorem frege67c 44582

Description: Lemma for frege68c 44583. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
frege67c	⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑)))

Proof of Theorem frege67c

Step	Hyp	Ref	Expression
1		frege59c.a	. . 3 ⊢ 𝐴 ∈ 𝐵
2	1	frege58c 44573	. 2 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)
3		frege7 44460	. 2 ⊢ ((∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) → (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑))))
4	2, 3	ax-mp 5	1 ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565   ∈ wcel 2149  [wsbc 3753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-frege1 44442  ax-frege2 44443  ax-frege58b 44553

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-sbc 3754

This theorem is referenced by:  frege68c  44583