Theorem frege60c 43390

Description: Swap antecedents of frege58c 43388. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
frege60c	⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))

Proof of Theorem frege60c

Step	Hyp	Ref	Expression
1		frege59c.a	. . . 4 ⊢ 𝐴 ∈ 𝐵
2	1	frege58c 43388	. . 3 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)))
3		sbcim1 3825	. . . 4 ⊢ ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)))
4		sbcim1 3825	. . . 4 ⊢ ([𝐴 / 𝑥](𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
5	3, 4	syl6 35	. . 3 ⊢ ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))
6	2, 5	syl 17	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))
7		frege12 43280	. 2 ⊢ ((∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) → (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒))))
8	6, 7	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))

Syntax hints:   → wi 4  ∀wal 1531   ∈ wcel 2098  [wsbc 3768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-frege1 43257  ax-frege2 43258  ax-frege8 43276  ax-frege58b 43368

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-sbc 3769

This theorem is referenced by:  frege93  43423