Theorem frege55lem2c 45178

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 3458	. . 3 ⊢ 𝑥 ∈ V
2	1	frege54cor1c 45176	. 2 ⊢ [𝑥 / 𝑧]𝑧 = 𝑥
3		frege53c 45175	. 2 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥))
4	2, 3	ax-mp 5	1 ⊢ (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3454  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-frege8 45070  ax-frege52c 45149

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-sbc 3743  df-sn 4589

This theorem is used by: (None)