Theorem frege55lem2c 37231

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 3176	. . 3 ⊢ 𝑥 ∈ V
2	1	frege54cor1c 37229	. 2 ⊢ [𝑥 / 𝑧]𝑧 = 𝑥
3		frege53c 37228	. 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 1475  Vcvv 3173  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-frege8 37123  ax-frege52c 37202

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-sn 4126

This theorem is referenced by: (None)