Theorem frege55lem2c 39052

Description: Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege55lem2c	⊢ (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥)

Distinct variable group:   𝑥,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem frege55lem2c

Step	Hyp	Ref	Expression
1		vex 3418	. . 3 ⊢ 𝑥 ∈ V
2	1	frege54cor1c 39050	. 2 ⊢ [𝑥 / 𝑧]𝑧 = 𝑥
3		frege53c 39049	. 2 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥))
4	2, 3	ax-mp 5	1 ⊢ (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥)

Syntax hints:   → wi 4   = wceq 1658  Vcvv 3415  [wsbc 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2804  ax-frege8 38944  ax-frege52c 39023

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-v 3417  df-sbc 3664  df-sn 4399

This theorem is referenced by: (None)